bounding satellite ranging errors due to the … · i certify that i have read this thesis and that...

USING KRIGING TO BOUND SATELLITE RANGING ERRORS DUE TO THE IONOSPHERE

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Juan Blanch

December 2003

© Copyright 2004 Juan Blanch All Rights Reserved

ii

I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality as a dissertation for the degree of Doctor of Philosophy.

_______________________________________ Per Enge

(Principal Advisor) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality as a dissertation for the degree of Doctor of Philosophy.

_______________________________________ Claire Tomlin


_______________________________________ Sanjay Lall


_______________________________________ Todd Walter

Approved for the University Committee on Graduate Studies:

_______________________________________

iii

Abstract

The Global Positioning System (GPS) has the potential to become the primary navigational

aid for civilian aircraft. GPS, however, lacks a fundamental feature for a safety critical

system: it does not provide hard bounds on the position inaccuracy. Although most of the

time the accuracy is excellent, the position error can become very large without any

warning to the user. Consequently, satellite based augmentation systems (SBAS) for GPS

have been developed to provide corrections and hard bounds on the user errors. Among the

sources of error to GPS positioning, the ionosphere is the largest and least predictable. The

only ionospheric information available to the United State’s SBAS, termed the Wide Area

Augmentation System (WAAS), at any given time is a set of slant range delay

measurements taken at reference stations distributed across the continent. From this limited

and randomly scattered data, the master station must compute in real time an estimate of

the ionospheric delay and a hard bound valid for any user within range of the reference

stations. The variability of the ionospheric behavior and the stringent integrity

requirements have caused the confidence bounds corresponding to the ionosphere to be

very large in WAAS, on the order of 10 meters of delay. In the position domain, these

conservative ranging bounds translate into conservative bounds on vertical position error.

These position error bounds are called protection levels and take values of 30 to 50 meters.

Since these values fluctuate near the maximum tolerable, WAAS is not always available.

In order to increase the availability of WAAS, we need to decrease the confidence bounds

corresponding to ionospheric uncertainty while maintaining integrity.

iv

In this work, I present an ionospheric estimation algorithm based on kriging. I first

introduce a simple model of the Vertical Ionospheric Delay that captures both the

deterministic behavior and the random behavior of the ionosphere. Under this model, the

kriging method, a type of minimum mean square estimator adapted to spatial data, is

optimal. More importantly kriging provides an estimation variance at each location that

can be easily translated to a confidence bound. However, this method must be modified for

two reasons; first, the state of the ionosphere is unknown and can only be estimated

through real time measurements; second, because of bandwidth constraints, the user cannot

receive all the measurements which are needed to apply kriging. I will show how these two

obstacles can be overcome.

The algorithm presented here provides up to 40% reduction in the confidence bound

corresponding to the ionospheric delay. This reduction in the ionospheric error results in a

20% reduction in Vertical Protection Levels across the United States. This is achieved

without any changes in the hardware and without significantly increasing the complexity of

the algorithm.

v

Acknowledgments

First, I would like to thank my advisor Professor Per Enge, for giving me the opportunity to

pursue this research, and for the guidance he has given me in the course of my doctoral

program. His insight and contagious enthusiasm has been a source of inspiration through

these years. I would also like to thank Dr. Todd Walter, director of the WAAS Laboratory.

He has helped me gain not only a deep and wide knowledge but a way of conducting

research, and this with endless patience. Working with him has been a pleasure. I hope to

continue learning from him in the future.

During the course of my research, I have received the generous advice from several

professors at Stanford. Professor Paul Switzer introduced me to the field of spatial

statistics, Professor André Journel gave me valuable advice on the use of geostatistics and

Professor Ingram Olkin in the use of multivariate analysis. I am very grateful to them, as

they have made my work much easier. Also, I would like to thank Professor Claire Tomlin

and Professor Sanjay Lall for taking the time to read this thesis.

I would like to thank research associates and students at the Stanford WAAS laboratory for

their help, always offered generously: Sriram Rajagopal, Seebany Datta-Barua, Juyong

Do, Shau-Shiun Jan, Frank Bauregger, Eric Phelts, Sherman Lo, Denis Akos, Demoz

Gebre-Egziabher, Frédéric Bastide, and Matt Deland. My thanks are extended to Doug

Archdeacon, Sherann Ellsworth and Dana Parga. I also want to thank Fiona Walter for her

careful and thoughtful proofreading of this thesis. Also, I thank the Federal Aviation

vi

Administration for its financial support of the GPS Laboratory and my research in

particular.

Finally, I would like to thank my parents for the support and understanding they have given

me during these years, and my wife, Vanessa, without whom this thesis would probably not

have been possible.

vii

Table of Contents

USING KRIGING TO BOUND SATELLITE RANGING ERRORS DUE TO THE IONOSPHERE ....I

ABSTRACT ........................................................................................................................................................IV

ACKNOWLEDGMENTS.................................................................................................................................VI

TABLE OF CONTENTS ...............................................................................................................................VIII

LIST OF FIGURES ...........................................................................................................................................XI

GLOSSARY OF ACRONYMS.....................................................................................................................XIII

CHAPTER 1 ......................................................................................................................................................... 1

INTRODUCTION................................................................................................................................................... 1 1.1 GPS AND CIVIL AIRCRAFT NAVIGATION........................................................................................................ 2 1.2 SATELLITE BASED AUGMENTATION SYSTEMS .............................................................................................. 4 1.2.1 PSEUDORANGE ERRORS ............................................................................................................................. 5 1.2.2 WAAS ARCHITECTURE ............................................................................................................................... 6 1.2.3 PROTECTION LIMIT CALCULATION ............................................................................................................ 7 1.3 REAL TIME IONOSPHERIC CORRECTIONS .................................................................................................... 10 1.3.1 PREVIOUS WORK...................................................................................................................................... 10 1.3.2 CURRENT WAAS IONOSPHERIC ESTIMATION ALGORITHM ....................................................................... 12 1.4 CONTRIBUTIONS ......................................................................................................................................... 13 1.4.1 OUTLINE .................................................................................................................................................. 13

CHAPTER 2 ....................................................................................................................................................... 15

PROBLEM STATEMENT..................................................................................................................................... 15 2.1 IONOSPHERIC PHYSICS................................................................................................................................ 15 2.1.1 IONOSPHERE STRUCTURE ........................................................................................................................ 16 2.1.2 IONOSPHERE PROPAGATION .................................................................................................................... 16 2.2 GPS IONOSPHERIC DELAY MEASUREMENTS................................................................................................ 18 2.2.1 ERRORS AND BIASES................................................................................................................................ 19

viii

2.2.2 HARDWARE BIAS CALIBRATION............................................................................................................... 20 2.2.3 RECEIVER NOISE AND MULTIPATH........................................................................................................... 20 2.2.4 POST-PROCESSED IONOSPHERIC DATA .................................................................................................... 22 2.2.5 MEASUREMENT NOISE COVARIANCE ....................................................................................................... 22 2.3 WAAS REFERENCE STATIONS...................................................................................................................... 23 2.4 REQUIREMENTS OF THE ALGORITHM.......................................................................................................... 23 2.4.1 PROBABILITY OF HAZARDOUSLY MISLEADING INFORMATION................................................................ 23 2.5 CONSTRAINTS ............................................................................................................................................. 25 2.5.1 IONOSPHERIC CONSTRAINTS.................................................................................................................... 25 2.5.2 SYSTEM CONSTRAINTS ............................................................................................................................ 25

CHAPTER 3 ....................................................................................................................................................... 27

IONOSPHERIC DELAY STRUCTURE................................................................................................................... 27 3.1 THIN SHELL APPROXIMATION ..................................................................................................................... 28 3.2 IONOSPHERE SNAPSHOTS............................................................................................................................ 30 3.2.1 QUIET CONDITIONS .................................................................................................................................. 30 3.2.2 DISTURBED CONDITIONS.......................................................................................................................... 32 3.3 VERTICAL IONOSPHERIC DELAY MODEL..................................................................................................... 33 3.3.1 SCATTER PLOTS ....................................................................................................................................... 34 3.3.2 GAUSSIAN ASSUMPTION .......................................................................................................................... 37 3.3.3 THEORETICAL AND EXPERIMENTAL VARIOGRAM ................................................................................... 39 3.3.4 MODEL VARIOGRAM AND COVARIANCE.................................................................................................. 41 3.3.5 DECORRELATION DURING A STORMY DAY .............................................................................................. 43 3.4 IONOSPHERIC IRREGULARITIES................................................................................................................... 46 3.4.1 ISOLATED IRREGULARITIES ..................................................................................................................... 46 3.4.2 GRADIENTS .............................................................................................................................................. 47 3.5 TEMPORAL DECORRELATION...................................................................................................................... 48 3.6 CONCLUSION............................................................................................................................................... 50

CHAPTER 4 ....................................................................................................................................................... 52

IONOSPHERIC ESTIMATION ALGORITHM ......................................................................................................... 52 4.1 ALGORITHM ASSUMING KNOWLEDGE OF THE MODEL................................................................................ 53 4.1.1 UNBIASED LINEAR ESTIMATOR................................................................................................................ 54 4.1.2 ESTIMATION VARIANCE........................................................................................................................... 55 4.1.3 ERROR BOUND AND ESTIMATION VARIANCE........................................................................................... 56 4.1.4 FINDING THE OPTIMAL COEFFICIENTS: KRIGING EQUATIONS .................................................................. 56 4.1.5 KRIGING MAPS ......................................................................................................................................... 58 4.1.6 ESTIMATION VARIANCE DEPENDENCE ON MEASUREMENT NOISE ........................................................... 60 4.2 IONOSPHERIC MODEL INFERENCE............................................................................................................... 62 4.2.1 MOTIVATION............................................................................................................................................ 62 4.2.2 DE-TRENDING AND DECORRELATING THE MEASUREMENTS ................................................................... 63 4.2.3 PHMI FORMULA........................................................................................................................................ 65 4.2.4 CASE WITH NO MEASUREMENT NOISE ..................................................................................................... 67 4.2.5 GENERAL CASE (WITH MEASUREMENT NOISE) ........................................................................................ 70 4.2.6 EXAMPLE ................................................................................................................................................. 71 4.2.7 RATIO BETWEEN PROCESS NOISE AND MEASUREMENT NOISE................................................................. 74 4.2.8 ADDING A THRESHOLD ............................................................................................................................ 77 4.2.9 DETERMINATION OF THE INFLATION FACTOR.......................................................................................... 81 4.3 THE THREAT MODEL ................................................................................................................................... 82

CHAPTER 5 ....................................................................................................................................................... 84

ADAPTATION OF THE ALGORITHM TO THE WAAS MESSAGE......................................................................... 84 5.1 THE IONOSPHERIC GRID: USER ALGORITHM ............................................................................................... 86 5.2 USER ERROR BOUND ................................................................................................................................... 87

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5.3 GIVE COMPUTATION ................................................................................................................................... 88 5.3.1 PRINCIPLE ................................................................................................................................................ 88 5.3.2 DECOMPOSING THE ESTIMATION VARIANCE IN THREE TERMS ................................................................ 89 5.3.3 BOUNDING THE QUADRATIC TERM .......................................................................................................... 90 5.3.4 BOUNDING THE LAST TERM ..................................................................................................................... 92 5.3.5 GIVE COMPUTATION: SUMMARY ............................................................................................................. 94

CHAPTER 6 ....................................................................................................................................................... 96

PERFORMANCE ANALYSIS ............................................................................................................................... 96 6.1 INTEGRITY CHECK ...................................................................................................................................... 97 6.1.1 LEAVE-ONE-OUT CROSS-VALIDATION..................................................................................................... 98 6.1.2 DATA DEPRIVATION............................................................................................................................... 101 6.2 PERFORMANCE METRICS........................................................................................................................... 102 6.2.1 MAAST ................................................................................................................................................... 103 6.2.2 GRID IONOSPHERIC VERTICAL ERROR AND VPL ................................................................................... 104 6.3 RESULTS WITHOUT THREAT MODEL ......................................................................................................... 105 6.3.1 QUIET DAY............................................................................................................................................. 106 6.3.2 DISTURBED DAY .................................................................................................................................... 108 6.4 RESULTS WITH THREAT MODEL ................................................................................................................ 111 6.4.1 QUIET DAY............................................................................................................................................. 111 6.4.2 DISTURBED DAY .................................................................................................................................... 114 6.5 LOSS OF AVAILABILITY DUE TO THE MESSAGE......................................................................................... 116

CHAPTER 7 ..................................................................................................................................................... 118

CONCLUSION.................................................................................................................................................. 118 7.1 SUMMARY OF RESULTS............................................................................................................................. 118 7.1.1 VERTICAL IONOSPHERIC DELAY STRUCTURE ........................................................................................ 118 7.1.2 IONOSPHERIC ESTIMATION .................................................................................................................... 119 7.1.3 PERFORMANCE ...................................................................................................................................... 120 7.2 SUGGESTIONS FOR FUTURE WORK........................................................................................................... 120

APPENDIX A ................................................................................................................................................... 122

COMPLEMENTARY DEFINITIONS AND PROOFS................................................................................................ 122 A.1 GAUSSIAN OVERBOUND........................................................................................................................... 122 A.2 INEQUALITY IN SECTION 4.2 .................................................................................................................... 123 A.3 BOUNDING ADDITIONAL TERM IN GIVE ................................................................................................... 123 A.4 CORRECTION TO THE PHMI EQUATION..................................................................................................... 125

BIBLIOGRAPHY ............................................................................................................................................ 131

x

List of Figures

Figure 1.1: Vertical Alarm Limit and Horizontal Alarm Limit ..............................................3

Figure 1.2: Sources of error in GPS .........................................................................................5

Figure 1.3: WAAS architecture................................................................................................6

Figure 2.1: CNMP standard deviation ...................................................................................21

Figure 3.1: Thin shell approximation over CONUS..............................................................29

Figure 3.2: Vertical ionospheric delay on July 2, 2000 at 5:00 p.m. E.T .............................30

Figure 3.3: Vertical ionospheric delay on July 2, 2000 at 12:00 p.m. E.T ...........................31

Figure 3.4: Vertical ionospheric delay on July 15, 2000 at 3:20 p.m. E.T ...........................32

Figure 3.5: Vertical ionospheric delay on April 6, 2000 at 4:50 p.m. E.T ...........................33

Figure 3.6: Estimating the differences of residuals ...............................................................35

Figure 3.7: Scatter plot of residuals corresponding to July 2, 2000......................................36

Figure 3.8: Sigma containment plot for July 2, 2000 ............................................................38

Figure 3.9: Variograms of the residual vertical ionospheric delay on July 2, 2000 .............40

Figure 3.10: Experimental variogram and model variogram ...............................................43

Figure 3.11: Sigma containment plots during April 6,2000..................................................44

Figure 3.12: Sigma containment plots during September 8, 2002........................................45

Figure 3.13: Very large gradient during the July 15-16, 2000 ionospheric storm................48

Figure 3.14: Spatial and temporal decorrelation during July 2, 2000...................................50

Figure 4.1: Map of the estimation variance on July 2, 2000 at 12:00 pm E.T.....................59

Figure 4.2: Histogram of the ratio σmeas2/ σprocess

2 on July 2, 2000 ........................................61

Figure 4.3: PHMI as a function of u parameterized by β ......................................................73

xi

Figure 4.4: Process noise contribution ...................................................................................76

Figure 4.5: PHMI as a function of u when a threshold is added ...........................................80

Figure 4.6: Real time inflation factor determination ............................................................82

Figure 5.1: Ionospheric grid points as specified in the MOPS..............................................85

Figure 5.2: Difference between kriging estimate and user estimate .....................................86

Figure 5.3: GIVE computation...............................................................................................89

Figure 5.4: Bounding the definite negative term ...................................................................91

Figure 5.5: Diagram of the covariance and a concave approximation..................................93

Figure 5.6: GIVE computation flowchart ..............................................................................95

Figure 6.1 Histogram of normalized residuals with single cross-validation.........................99

Figure 6.2: Q-Q plot of the normalized residuals ................................................................100

Figure 6.3 Histogram of normalized residuals with station removal ..................................101

Figure 6.4: Q-Q plot of the normalized residuals with station removal..............................102

Figure 6.5: GIVE map over CONUS...................................................................................105

Figure 6.6: GIVE map for July 2, 2000 using kriging.........................................................106

Figure 6.7: VPL map for July 2, 2000 using kriging...........................................................107

Figure 6.8: VPL map for July 2, 2000 using Raytheon’s algorithm...................................108

Figure 6.9: GIVE map for September 8, 2002 using kriging ..............................................109

Figure 6.10: VPL map for September 8, 2002 using kriging ..............................................110

Figure 6.11: VPL map for September 8, 2002 using Raytheon’s algorithm ......................110

Figure 6.12: GIVE map for July 2, 2000 using kriging with threat model.........................112

Figure 6.13: VPL map for July 2, 2000 using kriging with threat model ...........................113

Figure 6.14: VPL map for July 2, 2000 using Raytheon’s algorithm with threat model ...113

Figure 6.15: GIVE map for September 8, 2002 using kriging............................................114

Figure 6.16: VPL map for September 8, 2002 using kriging ..............................................115

Figure 6.17: VPL map for September 8, 2002 using Raytheon’s algorithm ......................115

Figure 6.18: VPL map for July 2, 2000 using kriging without bandwidth limitations.......116

Figure 6.19: VPL map for July 2, 2000 using kriging without quantization ......................117

xii

Glossary of Acronyms

Wide-Area DGPS

CNMP Code Noise and Multipath

CONUS Conterminous United States

GIVE Grid Ionospheric Vertical Error

HAL Horizontal Alert Limit

HMI Hazardously Misleading Information

HPL Horizontal Protection Level

IFB Interfrequency bias

IGP Ionospheric Grid Point

IPP Ionospheric Pierce Point

MOPS Minimum Operational Performance Standard. Refers to RTCA159

specification for the Wide-Area Augmentation System (WAAS).

PHMI Probability of Hazardously Misleading Information

Protection Level

Broadcast indication of the bound on the accuracy of the state. This value is

compared to the Alert Limit to determine if a flight operation can continue.

SBAS Space Based Augmentation System

TEC Total Electron Content

UDRE User Differential Range Error

xiii

VAL Vertical Alert Limit

VPL Vertical Protection Level

WAAS Wide-Area Augmentation System

WMS WAAS Master Station

WRS WAAS Reference Station

xiv

Chapter 1

Introduction

The largest obstacles for the Global Positioning System (GPS) to become the primary

navigational aid in civil aircraft are the ionosphere [Chao] and radio frequency interference.

Ionospheric disturbances can cause positioning errors larger than 50 meters. In this thesis,

we present an algorithm that, from a set of measurements of the ionosphere, corrects most

of the error induced by the ionosphere and lets the user know the bound on the inaccuracy

of the correction. The method used is based on kriging [Cressie], [Journel], a type of

minimum mean square estimator adapted to spatial data. However, many changes were

necessary to adapt it to this particular problem.

In this chapter, we will first introduce some background on GPS, aircraft navigation, and

the need for GPS augmentation systems. This will be followed by a description of satellite

based augmentation systems and how they enable GPS to become a certified navigational

aid. We will then focus on the ionospheric corrections broadcast by the augmentation

systems and present previous work relevant to this thesis. Finally, an outline of the thesis

will be given.

INTRODUCTION 2 1.1 GPS AND CIVIL AIRCRAFT NAVIGATION

GPS is a positioning system based on ranging. Each of the 24 GPS satellites in medium

earth orbit broadcasts a message that includes the position of the satellite and the time it

was sent. The message is broadcast at the frequencies L1 (1575.42 MHz) and L2 (1227.60

MHz) and it is modulated by a pseudorandom (PRN) code, such that all satellites can

broadcast at the same frequencies without time sharing [Enge]. The frequency L1 is public

whereas L2 is encrypted so that only authorized users can take full advantage of it. A user

measures the apparent distance to the satellite by comparing the time each message was

sent with the current time. This quantity is called pseudorange. By combining the

pseudoranges appropriately, users can find their position. Using stand alone GPS (L1

only), which is available to anyone, the horizontal position error is rarely above 10 meters

[Enge01]. For this reason, the potential of GPS for civil aircraft navigation was recognized

very early: never before was there a navigational aid so ubiquitous and precise at the same

time.

However, for aircraft navigation, it is not enough to be precise most of the time. What is

needed is the maximum possible position error: pilots want to know that the error is not

going to be larger than a certain bound (called the Vertical Alarm Limit (VAL) in the

vertical direction and the Horizontal Alarm Limit (HAL) in the horizontal). Figure 1.1

illustrates this point. If there is no guarantee that the position lies within the red box, the

pilot will not be able to go between the two mountains. As a consequence, a useful

navigation aid must notify within a certain time (time to alarm) that the positioning error

might be larger than the VAL and HAL. This property is called integrity: the system must

always be believable.

INTRODUCTION 3

Figure 1.1: Vertical Alarm Limit and Horizontal Alarm Limit

GPS by itself does not have sufficient integrity. There is no mechanism to compute a hard

bound on the position error. Although the errors are typically very small, they can become

very large without any warning to the user. As we will see later in this chapter, several

sources of error can corrupt the pseudorange measurement. These errors in the

pseudorange have a direct effect on the position error. GPS by itself cannot therefore be

used as a navigational aid for common civil aviation. To fill this gap, the concept of the

augmentation system was introduced [Braff]. (Of course, all existing aids to air navigation

have monitors, so it is only a continuation of this practice with the newest aid to

navigation.) The idea is to monitor GPS signals through reference stations and send

corrections and integrity flags to the user so that: first, the position error is reduced;

second, the user can determine a hard bound on the position error. There are two concepts

for augmentation systems: ground based and satellite based. Ground based augmentation

systems are local and only need one reference station [Chou]. (Actually, there are several

reference stations that check on each other and are collocated.) Since most of the errors

have a strong spatial correlation and the position of the reference receiver is known, it is

possible to correct most of the user position error. Satellite based augmentation systems

(SBAS) operate in a different manner: the sources of error in the pseudoranges are isolated

INTRODUCTION 4 and corrected one by one [Kee]. This concept requires several well equipped reference

stations and a satellite that can broadcast the correction message. To this date the Wide

Area Augmentation System (WAAS) is the only SBAS in operation. WAAS was

commissioned by the FAA and started operations over a large part of CONUS on July 10,

2003. There are several other SBAS systems under development: the European GPS

Overlay System in Europe [Benedicto], MSAS (MTSAT Satellite-Based Augmentation

System) in Japan [Shimamura], GAGAN in India, and several other projects at an earlier

stage.

1.2 SATELLITE BASED AUGMENTATION SYSTEMS

As said earlier, the purpose of satellite based augmentation systems is to allow users to

correct their position error as much as possible and, more importantly, to generate a hard

bound on the error over a large geographic area. This hard bound is called the protection

level. In the vertical, the hard bound is labeled Vertical Protection Level (VPL) and in the

horizontal, Horizontal Protection Level (HPL). More precisely, a protection limit is

defined such that at any moment:

( ) 7Prob 10error PL −< ≤

For an augmentation system to be useful, we need the protection levels to be smaller than

the VAL and HAL corresponding to the maneuver that is underway.

In Subsection 1.2.1 we will list the disturbances to which the pseudoranges are subject and

distinguish which ones can be corrected by an augmentation system. In Subsection 1.2.2,

the overall architecture of WAAS is described (the architecture of all the other satellite

based augmentation systems is identical). Finally, we explain how the user calculates his

VPL and HPL based on the information contained in the WAAS message.

INTRODUCTION 5 1.2.1 PSEUDORANGE ERRORS

Clock errors Ephemeris errors

Ionospheric delay

Tropospheric delay

Multipath

Receiver noise

Figure 1.2: Sources of error in GPS

Figure 1.2 illustrates the different sources of errors affecting the pseudoranges for a single

frequency user. As we said earlier, the GPS message includes the satellite position and a

time tag indicating the time the message was sent. Going from the satellite to the user, the

first possible set of errors is due to the ephemeris error (difference between true position of

the satellite and broadcast position) and the clock error (difference between the true clock

and the satellite clock) [Enge01]. The second error is introduced by the ionosphere. As we

will detail later, the ionosphere introduces an additional delay in the message that depends

on the free electron content of the upper atmosphere. Because this error is frequency

dependent, dual frequency users can correct this error without help. However, single

frequency users have no means to measure the additional delay introduced by the

ionosphere. The third possible error is introduced by the troposphere, which also

introduces an additional delay. This delay is not dispersive, so it cannot be extracted using

dual frequency. Fortunately, it can be corrected using relatively simple models [Spilker].

The remaining errors are multipath and receiver noise. Multipath is due to the possible

INTRODUCTION 6 transmission of the message through paths others than line of sight. These additional paths

(which usually have lower power) can distort the correlation peak [Braasch] such that the

measured pseudorange is either larger or smaller. The receiver noise is due to the

limitations inherent to the receiver. In this thesis, we will explain the origin of the

ionospheric delay. Also, because it is relevant to ionospheric estimation, we will see in

Chapter 2 how multipath and receiver noise are characterized.

1.2.2 WAAS ARCHITECTURE

The architecture of WAAS is summarized in Figure 1.3. The GPS signals are received by a

network of WAAS reference stations (WRS) located in the Conterminous United States

(CONUS) and in neighboring regions. The GPS measurements are sent to be processed by

the master station. Because the location of the WRS is known and the receivers have dual

frequency, it is possible to isolate some of the errors in the pseudorange.

Figure 1.3: WAAS architecture

GPS message Geo. Uplink

• Ranging source

• Error bounds

• Corrections

Master Station

INTRODUCTION 7 There are three types of errors that can be corrected – to a certain extent, as we will see -;

the ephemeris error and the clock error for each of the satellites, and the ionospheric delay.

The master station produces a message – the WAAS message – that contains integrity

information and real time corrections for the ephemeris and clock error for each satellite

along with the ionospheric delay. The message is continually uploaded to a geostationary

satellite. The geostationary satellite broadcasts the message to the users at 250 bits per

second in a PRN code on L1 (1575.42 MHz). In addition to the WAAS message, the

geostationary satellite is a valuable source of ranging. For a more detailed introduction to

the WAAS architecture we refer the reader to [Enge96].

A WAAS user first measures the pseudoranges to the GPS satellites in view and to the

geostationary satellite. Then, using the information contained in the WAAS message it

determines which satellites can be used safely, and corrects their pseudoranges (ephemeris

and clock error and ionospheric delay). Also, the user computes a VPL and an HPL.

These are then compared to the VAL and HAL to determine if the system can be used for

the desired procedure.

1.2.3 PROTECTION LIMIT CALCULATION

In this subsection we give a brief description of the Protection Level calculations. Let us

first review the equations used in WAAS [Walter97]. The equations to solve are:

( ) ( ) ( )k kc x x b k

ρρ ε= − + +

In this equation, ( )kcρ is the pseudorange after the WAAS correction for the kth satellite,

ρε are the errors that remain after the correction and b is the user clock bias. After

linearization, and assuming that we have initial estimates of the position and the clock bias

x0 and b0, we have:

INTRODUCTION 8

00

0c

x x xG G

b b bρ ρ

δρ ρ δρ ε ε

δ−⎡ ⎤ ⎡ ⎤

− = + =⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦+

In these equations, cρ is the vector of pseudoranges after the WAAS correction; 0ρ is the

pseudoranges that would be measured if the user were located in x0 and its clock bias were

b0; G is a K x 4 matrix characterizing the user-satellite geometry (K is the number of valid

satellites in view):

( )( )( )( )

( )( )

1

2

1 1

1 1

1 1

T

T

TK

G

⎡ ⎤−⎢ ⎥⎢ ⎥−⎢ ⎥= ⎢ ⎥

⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

The first three columns are the components of the line of sight vector to each of the

satellites. ρε needs to be characterized in order to compute the Protection Limits. Let us

suppose now that the errors are Gaussian and independent. The covariance matrix of the

errors can be written:

21

21 2

2

0 00 0

00 0 0 K

W

σσ

σ

−

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

In these conditions, the optimal solution is given by weighted least squares:

( ) 1ˆˆ

T Tx

G WG G Wb

δδρ

δ−⎡ ⎤

=⎢ ⎥⎣ ⎦

(Even if the errors are non Gaussian, the mean squared error is minimized.) The vertical

error is then characterized by the standard deviation:

INTRODUCTION 9

( ) 12

3,3

TVerticalError G WGσ

−=

if the third component in x refers to the height. Assuming Gaussian errors, the VPL can be

computed as:

,V PA VerticalErrorVPL K σ=

Here, KV,PA is the quantile of a unit Gaussian distribution corresponding to 10-7 and is equal

to 5.33. There is a difficulty here: the errors are not Gaussian and not necessarily

uncorrelated. The approach taken in SBAS is, for each pseudorange error, to allow the user

to compute a Gaussian overbound of the true error distribution assuming that the different

errors are uncorrelated. The definition of Gaussian overbound is given in Appendix A, but

for now it is enough to see it as a Gaussian distribution that conservatively describes the

true distribution. Ignoring the correlation tends to be a conservative step, in the sense that

it leads to larger protection limits [Walter97]. When it is not a conservative step, WAAS

inflates the individual error bounds to account for the effect of the correlation [Schempp].

The validity of this approach has been confirmed by several years of experience first in the

WAAS test bed (the NSTB network) [Dehel] and then in the operational WAAS

[McHugh].

As said earlier, the WAAS message allows the user to form the matrix W. For each

satellite we have [MOPS]:

2 2 2 2 2, , , ,i i flt i UIRE i receiver i tropoσ σ σ σ σ= + + +

The two last terms are provided by the user. The third one, labeled i, receiver, is the

receiver noise and multipath overbound. The fourth term, labeled i,tropo, is the delay

introduced by the troposphere. The first term, labeled flt (fast and long term) includes the

User Differential Range Error, which is the error due the ephemeris and clock error. The

information needed to compute this term is included in the WAAS message.

INTRODUCTION 10

The second term, , the User Ionospheric Range Error, is also computed using the

WAAS message. This term is an overbound of the error remaining in the range delay once

the ionospheric correction has been applied. The goal of this thesis is to make this term as

small as possible while maintaining the integrity of the system.

2,i UIREσ

1.3 REAL TIME IONOSPHERIC CORRECTIONS

As said earlier, the ionospheric delay corrections and the protection limits are computed in

real time from the measurements collected at the reference stations. This is possible

because of the strong spatial correlation of the ionospheric delay. (In Chapter 3, this

correlation will be stated explicitly.) However, the degree of correlation is quite variable

and must be monitored in real time. This variability forces the error bound on the

ionospheric delay to be large. The problem is made harder by the fact that the available

ionospheric measurements are sparse and subject to measurement noise. All of this has

made the ionospheric correction term the dominant term in the VPL.

In this section, previous research on ionospheric estimation is presented. In the first

subsection we cite ionospheric modeling and general estimation (not necessarily for

navigation use.) Since there is a large amount of previous work in this field, we will only

cite the works most relevant to this thesis. The second subsection presents work

specifically addressing ionospheric estimation with an emphasis on integrity.

1.3.1 PREVIOUS WORK

The first class of previous work deals with ionospheric mapping. The goal of these studies

is to find consistent representations of the ionosphere given the available measurements in

order to draw conclusions about what is happening in the ionosphere.

To our knowledge, the first attempts to characterize the random properties of the

ionospheric delay as measured by GPS were made by Delikaraoglou [Delika] and Wild

[Wild]. In those early works, there are several interesting points relevant to this thesis.

INTRODUCTION 11 First, temporal and spatial autocorrelation plots are computed from GPS ionospheric

measurements. Second, the difference between the stochastic behavior of the ionosphere

and the measurement noise is clearly drawn. Third, it is recognized that, within the thin

shell model (which will be explained in Chapter 3) the ionospheric delay can be interpreted

as having a deterministic component and a stochastic component. Finally, an estimation

method close to kriging, called co-location, is presented (the difference with kriging, is that

the determination of the trend is done visually, which prevents it from being made

automatic). The application of kriging to ionospheric mapping was introduced by P.A.

Bradley in [Bradley]. These techniques have been applied to the mapping of different

ionospheric characteristics, in particular the Total Electron Content (TEC) by Stanislawska

et al [Stanis]. For our purpose, what is lacking in these studies is the computation of a

reliable error bound on the estimate.

Another class of ionospheric estimation algorithms is based on tomography. Because the

ionosphere is three-dimensional and the GPS measurements are integrals along the ray

path, it is natural to try to reconstruct the Total Electron Content using tomography. The

most relevant effort to this research has been realized by A. Hansen [Hansen02]. In this

same reference, there is an overview of all ionospheric estimation algorithms based on

tomography. Compared to previous applications of tomography to ionosphere estimation,

there is an attempt to produce a tight error bound on the estimate. In addition, this

ionospheric estimation algorithm was adapted to WAAS.

Previous work on ionospheric estimation specific to satellite based augmentation systems

includes all the early work done for WAAS [Chao]. At that point, the ionosphere was

believed to be sufficiently well behaved so that a single model for ionospheric behavior

would be enough. Error bounds were computed that depended on the geometry of the

measurements, but they assumed smooth ionospheric models corresponding to low

ionospheric activity.

INTRODUCTION 12 1.3.2 CURRENT WAAS IONOSPHERIC ESTIMATION ALGORITHM

The previous work that is the most relevant to this thesis is all the effort that has led to the

current ionospheric estimation algorithm for WAAS. In 1999, at the beginning of the solar

maximum season (one of the periods in the 11 year solar cycle [Hargreaves]) it was found

that assuming smooth ionospheric behavior during ionospheric storms could lead to

integrity faults in WAAS.

The difference between quiet and stormy ionospheric behavior was exposed in [Hansen00]

through decorrelation plots. The large difference between quiet and stormy ionospheric

behavior resulted in the development of the storm detector [Walter00], which is the basis of

the current WAAS ionospheric algorithm. It is interesting to summarize this paper here, as

several ideas from it are used in this thesis. The idea of the algorithm is to test whether the

real time measurements (at a given time) are compatible with a quiet ionosphere via a chi-

square test on the measurements. If the chi-square test statistic is larger than a certain

threshold, then the ionosphere is assumed to be stormy, no correction is attempted, and a

very large error bound is set, which will virtually cover any possible error, but will make

the system unavailable for precision approach. If the chi-square statistic is below the

threshold, then the ionosphere is probably in a quiet state. However, the error bound is

inflated because even if the statistic is low, there is still a possibility that the ionosphere is

mildly disturbed. Improvements to this algorithm have been suggested, including but not

limited to the use of a dynamic inflation factor [Angus], [Cormier].

The previous algorithm was designed assuming that the ionosphere was well sampled by

the WAAS reference station measurements. Unfortunately, while this is true in most of the

CONUS region, it does not hold at the edge of coverage. There, the measurements become

sparse as there cannot be reference stations in the ocean. The concern is that the

ionosphere might appear quiet in the WAAS measurements when, in reality, there are

disturbances very close to the measurements that are not detected. This concern led to the

development of the threat model [Sparks01]. The threat model is a methodology to put a

hard bound on the possible threats present in archived real ionospheric data. A threat, here,

is defined as a class of ionospheric behavior that could lead to an integrity failure. As it is

INTRODUCTION 13 an essential part of the algorithm that is reused in this thesis, it will be presented with some

details in Chapter 4. Further improvements of the ionospheric threat model have been

proposed in [Sparks03].

1.4 CONTRIBUTIONS

Since many of the contributions need a long introduction which will be given in the thesis,

we will only give a brief summary of them here. We found a simple local characterization

of the vertical ionospheric delay as a deterministic component and random field correlated

with distance; this is a refinement of previous work. To our knowledge, we have

introduced the first application of kriging in a safety critical system and in real time (both

are new). To do that, we evaluated the effect of measurement noise on the determination of

the underlying random model. This is one of the most valuable contributions of this thesis

as it can be adapted to other critical systems where there is both process noise and

measurement noise but where the characteristics of the process noise are not well known.

The last important contribution resides in the fact that the ionospheric estimation algorithm

designed in this thesis could provide a 20% reduction in VPL compared to the current

algorithm, with the same safety standard and without changing the current SBAS message

standards.

1.4.1 OUTLINE

The second chapter gives a more detailed picture of the ionosphere and of the available

sensors for estimation, an explanation of the algorithm requirements and the problem

statement. The third chapter explains how to obtain a simple and useful model of the

vertical ionospheric delay and how to reduce the degree of disturbance to a single

parameter; this chapter also shows some of the limits of the model. The fourth chapter uses

the model derived in the third chapter to derive an estimation algorithm. There are two

different sections each one dealing with a different side of the problem: irregularly

scattered measurements and uncertain underlying random model. At the end of Chapter 4

INTRODUCTION 14 we include a description of the threat model. It is not a contribution in this thesis, but it

needs to be explained since it is a required part of the algorithm. Chapter 5 shows how to

modify the algorithm designed in Chapter 4 to the current SBAS standard. Finally, in

Chapter 6 we evaluate the algorithm using a Service Volume Analysis tool.

Chapter 2

Problem Statement

The purpose of this chapter is to define the problems. What is the effect of the ionosphere

on the GPS signal? How are the GPS measurements processed to extract ionosphere

information? What are the requirements on the ionosphere estimation algorithm? First, we

will give a brief description of the ionosphere and its effects on radio wave propagation.

We will then give the formula that links the electron content with the range delay

experienced by a user. From there, we will see how the dispersive nature of the ionosphere

allows a dual frequency receiver to compute the range delay introduced by the ionosphere.

We will review how the GPS measurements are processed, and how their noise is

characterized. Then we will turn our attention to the specific requirements of the

ionospheric estimation algorithm for satellite navigation. As this chapter is an overview of

previous work, it will be brief. Equation Section 2

2.1 IONOSPHERIC PHYSICS

We start by giving a brief description of the ionosphere and its effect on GPS

measurements, and refer the reader to several references that cover the subject extensively.

For the description of the ionosphere, the focus is on the Total Electron Content (which will

be defined.)

PROBLEM STATEMENT 16 2.1.1 IONOSPHERE STRUCTURE

The ionosphere is a region of the upper atmosphere constituted of partially ionized plasma.

This region extends from around 100 km height to above 1000 km. The ionization results

from the ultraviolet radiation from the Sun. The exact electron density at any location is

caused by a complex interplay between the solar flux and the terrestrial magnetic field.

Because the UV radiation is not constant and the earth is rotating, the electron density is

variable both spatially and temporally. There is a diurnal pattern, where the maximum

density is located near the equator and lags the sun by two to three hours. From that point,

the electron density tapers off toward the poles. Despite the gross regularity, the details are

variable and difficult to predict. We will not discuss the several models that have been

developed to integrate all physical phenomena affecting the ionosphere and predict the

electron density. What is important for us here is that none of them has been very

successful in predicting this day-to-day variability [Klobuchar].

Because the electron density is a function of the solar flux, it is linked to the 11 year solar

cycle. During this cycle the UV flux goes through a maximum. During this period, the

ionosphere is more likely to be disturbed, in the sense that the electron density can become

much higher and much more unpredictable than on a quiet day. Also, the daily pattern is

disturbed during these stormy events. Although more likely during the high solar season,

these events can happen during “non-peak” portions of the solar cycle.

2.1.2 IONOSPHERE PROPAGATION

The major effects of the ionosphere on the GPS signal are: Doppler shift, Faraday rotation

of linearly polarized signals, bending of the radio wave, scintillation and delay. Here we

will only discuss the delay introduced by the ionosphere on the GPS signal, as it is by far

the most severe effect in mid-latitudes (in Equatorial and high latitudes, scintillation is also

a problem). The ionosphere is a dispersive medium: the refractive index is dependent on

the frequency. More precisely, we have the approximation (obtained from the Appleton

and Hartree general equation for the refractive index of the ionosphere [Hargreaves]):

PROBLEM STATEMENT 17

( ) ( ) ( )2

2 20

40.3, 1 12 2

ee

N r en r f N r

mf fε= − = − (2.1)

In this equation, is the refractive index at location r, f is the frequency, and

is the electron density.

( ,n r f )

( )eN r

This equation allows us to evaluate the effect of the ionosphere on the GPS signal. The

group delay (the delay suffered by the message) of a ray path crossing the ionosphere is

given by:

( )1 n dlρ∆ = −∫

By plugging Equation (2.1) into this last equation we get:

22

40.3 (in number of electrons per m )eN dlf

ρ∆ = ∫ (2.2)

This last equation shows that the group delay is proportional to the inverse of the frequency

squared and to the quantity which is called the Total Electron Content (TEC) and it

is expressed in number of electrons per square meter.

eN dl∫

In addition to the group delay, the refractive index provokes an advance of the carrier phase

[Klobuchar]. The carrier phase ∆Φ advance is such that:

λ ρ∆Φ = −∆ (2.3)

Equations (2.2) and (2.3) express the two effects that we are going to try to mitigate in this

thesis. A user tracking the GPS signal at only one frequency cannot observe and correct

the delay introduced by the ionosphere. This is the reason an ionospheric estimation

algorithm is needed. (It is possible to detect relative changes in the ionosphere by

comparing the carrier phase and the code delay [Klobuchar]. This possibility is not

exploited in satellite based augmentation systems because it would put more responsibility

PROBLEM STATEMENT 18 on the user side than is desirable, and would not be robust enough, because the bias is

poorly observed [Enge96]).)

2.2 GPS IONOSPHERIC DELAY MEASUREMENTS

For a single frequency user, the delay (2.2) is not observable. However, using two different

frequencies, the delay (2.2) can be isolated. GPS transmits its message in two frequencies:

L1 (1575.42 MHz) and L2 (1227.6 MHz). Excluding all the other errors, two pseudorange

measurements from the same receiver to the same satellite differ by:

( )1 2 1

240.3 1L L Lf TECρ ρ γ− = −

where 1

2

2

2L

L

ff

γ = . The delay in meters at L1 is then (we can keep the notation TEC for the

delay since it is proportional to the TEC in electrons per square meter):

( )1 2

1L LTEC

ρ ργ−

=−

Similarly, we have for the carrier phase measurements:

( )1 2

1L LTECγ

∆Φ −∆Φ= −

−

While the code measurement is absolute and noisy, the carrier phase measurement has a

much lower noise, but is ambiguous (in the sense that the measurement is accurate modulo

the length of a wave. In the next section, this ambiguity is expressed through the unknown

Ni) These two estimates can be combined to form an improved TEC estimate. We now go

into more detail.

PROBLEM STATEMENT 19 2.2.1 ERRORS AND BIASES

The situation is not as simple as the previous equations suggest. There are many more

terms in the pseudorange measurement that impair our observability of the ionospheric

delay. The four available measurements are:

1 1L Lpr R TEC M1Lξ= + + +

( )2 2L gdpr R TEC IFB M

2L Lγ τ ξ= + + + + +

1 1 1L R M TEC Nφ φφ ξ λ= + + − +

( )2 2 2L gR M TEC IFB Nφ φ dφ ξ γ τ λ= + + − − − +

In these equations, R designates the range and the clock biases (satellite and receiver). This

term is common to all four measurements. M is the multipath (error due to the fact that not

all the received signals are from the line of sight [Braasch]). The multipath term is

negligible in the carrier phase measurements. ξ is the receiver noise; it is different for each

of the terms and also negligible in the carrier phase measurements. IFB is the

interfrequency bias in the receiver, and τgd is the interfrequency bias in the satellite. Ni is

the ambiguity in the carrier phase measurement and is unknown. Neglecting multipath and

receiver noise in the carrier phase measurement we get the two observables:

( ) ( )2 1 1

1ˆ1 1code gd L L L LTEC TEC IFB M M

2

γ τ ξ ξγ γ

= + + + − + −− −

( ) ( )2 2 1 11ˆ

1 1phase gdTEC TEC IFB N Nγ τ λ λγ γ

= + + − −− −

These two measurements are combined to provide a low noise estimate of the ionospheric

delay. Second, we want to characterize our measurement uncertainty in a conservative

way. To do that we need to calibrate the biases – which have a time constant of days - and

get a bound on the receiver noise and multipath.

PROBLEM STATEMENT 20 2.2.2 HARDWARE BIAS CALIBRATION

The hardware bias calibration (IFB and group delay) is done using software based on the

Global Ionospheric Model [Wilson]. The idea is to assume that the ionosphere is stationary

in a frame moving with the sun along the magnetic equator. The ionosphere is then

described by a set of coefficients representing the magnitude of the electron density

function. The problem is then to estimate the hardware bias and the coefficients at the

same time from the observed measurements. There are two versions of the hardware

calibration: a batch solution (used for post-processing) and a sequential solution. Because

the biases evolve very slowly (on the order of days) the bias estimation algorithm can use

long time constants (for the sequential filter) or several days worth of ionospheric data (for

the batch solution). The equations describing the relationship between the coefficients,

biases and measurements are linear, so the problem can be solved efficiently using least

squares in the batch version of the estimator and a Kalman filter in the case of the real time

estimator.

It has been shown that the bias determination is not very sensitive to the ionospheric model

that is assumed [Hansen02], and that departures from the ionospheric model do not harm

the accuracy of the bias estimates. The biases can be calibrated to better than 10 cm and

errors in the biases do not exceed 1 meter.

2.2.3 RECEIVER NOISE AND MULTIPATH

There are several techniques to mitigate and characterize the receiver noise and multipath,

also labeled code noise and multipath (CNMP). The technique that was selected for

WAAS estimates and corrects for the multipath and then provides a noise estimate for the

residual error in the corrected measurements. This approach is detailed in [Graas]. A high

level description is provided in [Shallberg]. In this section we summarize the account

given in [Shallberg].

The CNMP algorithm has three major steps: a mean filter, a mean error function, and cycle

slip detection. Multipath error is calculated using the standard CNMP relationship for dual

PROBLEM STATEMENT 21 frequency code pseudorange and the carrier phase (which can be found in [Shallberg].)

Due to the ambiguities inherent to the carrier phase measurement, the estimate of the

multipath error is biased. The bias is removed using the mean filter estimator. The final

multipath correction is obtained by differencing the mean filter estimate from the current

multipath estimate. Finally, the multipath corrected pseudoranges are carrier smoothed

with a period equal to the mean filter time constant. The mean error function is the residual

error once the estimate of the multipath has been removed. Figure 2.1 shows the

overbound of the mean error function as a function of time.

0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

1.5

2

2.5

3

Time of continuous l1 and L2 tracking in seconds

CN

MP

sta

ndar

d de

viat

ion

in m

eter

s

Figure 2.1: CNMP standard deviation

Because of the mean filter, the multipath correction estimate is not accurate at the

beginning of the satellite track. However, as time passes, the estimate becomes

increasingly accurate due to the more accurate determination of the carrier ambiguities.

The mean error function was determined empirically making use of WAAS measurements.

PROBLEM STATEMENT 22 It represents the standard deviation of a Gaussian overbound (see Appendix A) of the

residual error. The mean error function starts out large and decreases towards a floor value

that is reached after 12000 s. The last piece of the CNMP algorithm is the cycle slip

detector. It uses both a dual frequency carrier phase detector and a single frequency carrier

phase detector. The single frequency detector is there to rule out any simultaneous cycle

slips in L1 and L2 that would be unobservable for the dual frequency detector. Cycle slips

force data gaps to restart the algorithm.

2.2.4 POST-PROCESSED IONOSPHERIC DATA

In the previous sections we have dealt with the real time measurements that will be fed into

the ionospheric estimation algorithm. We now turn our attention to post-processed data.

Post-processing raw data allows us to remove noise that we would not be able to remove in

real time, and thus obtain very accurate ionospheric delay measurements. A description of

the processing can be found in [Hansen00]. As we will see in Chapter 3, it is critical to

have low measurement noise and an extensive data set of ionospheric delay measurements

to be able to correctly model the ionosphere. This is all the more important for WAAS, as

it has been decided early on that the design of the algorithm and the safety analysis would

rely on measurements. Here, we outline each step to produce a ‘supertruth’ data set from

raw measurements collected at the WAAS reference stations. For more details, please refer

to [Hansen00].

2.2.5 MEASUREMENT NOISE COVARIANCE

None of the calibration and data processing described above is unique to this thesis. What

is important here, is that after this process (be it real time or post-processed) we end up

with a set of measurement data at each time frame. The accuracy of the measurements is

characterized by a covariance matrix M, which characterizes both the code noise and

multipath error (after correction) and the uncertainty introduced by the bias (after

correction also). This matrix is almost diagonal. There are off-diagonal terms because the

error in the bias estimation is common to all measurements coming from the same satellite

or the same reference station. For every measurement, the standard deviation (the diagonal

PROBLEM STATEMENT 23 element corresponding to the measurement) is a Gaussian overbound of the true error

distribution. The covariance M naturally varies with time and the magnitude of the

standard deviation of the errors is on the order of 50 cm.

2.3 WAAS REFERENCE STATIONS

The measurements described earlier are collected through a network of reference stations.

The location of these reference stations can be seen in Figure 1.3. Each reference station

has several dual frequency capable receivers, a precise clock and an antenna located in a

low multipath environment. Several dual frequency receivers provide diversity in the

measurements. This diversity reduces the impact of noise and aids the detection and

isolation of faults in every single measurement.

2.4 REQUIREMENTS OF THE ALGORITHM

As discussed in Chapter 1, the purpose of WAAS (and all SBAS systems) is to allow the

user to compute a hard bound on the position error (VPL or HPL) or to let the user know

that it is not safe to use a certain measurement (because the uncertainty on the error is too

large). We will first present the exact requirement on the ionospheric estimation algorithm

and then describe why it is difficult to achieve.

2.4.1 PROBABILITY OF HAZARDOUSLY MISLEADING INFORMATION

As we said in the Introduction, the user must receive a correction and Gaussian overbound

for each of the errors, or a flag indicating that the pseudorange cannot be used safely

[MOPS]. For the ionospheric delay, when a valid correction is sent (along with an error

bound), the Gaussian overbound condition can be written:

( ) ( ) for all 0est true estP I I K Q K Kσ− ≥ ≤ ≥ (2.4)

PROBLEM STATEMENT 24 In this equation, estI is the ionospheric delay computed by the user, trueI is the real

ionospheric delay, and Q is the residual of the normal cumulative distribution function.

estσ is the standard deviation of the Gaussian overbound. We now describe how Condition

(2.4) can be simplified. First of all, the Gaussian overbound theorem (See Appendix A and

[DeCleene]) can be relaxed in our case, because we only need a VPL such that:

( ) 7, 3,3vertical error < 10V PAP K d −<

As a result, we only need to meet Condition (2.4) for K in a finite range. In this thesis we

will assume that it is enough to satisfy (2.4) when K<5.592. We can further simplify the

requirement by noticing that it is easier to fulfill condition (2.4) using small K than large K.

In other words, the solution of the equation:

( ) ( )est true estP I I K Q Kσ− ≥ =

where estσ is the unknown, increases as a function of K. (We are not proving this now, but

it will be revisited in Chapter 4.) Consequently, it is enough to have:

( ) ( ) for 5.592est true estP I I K Q K Kσ− ≥ ≤ = (2.5)

A failure to fulfill Condition (2.5) constitutes what is called Hazardously Misleading

Information (HMI). The left side of Inequality (2.5) is called Probability of Hazardously

Misleading Information (PHMI).

The problem now is defined: a WAAS user needs to be able to compute an estimate of each

ionospheric delay corresponding to each satellite in view and an error bound such that (2.5)

is fulfilled at any moment. It would be easy to send a very large error bound, we would

then be sure that (2.5) is met. However, for WAAS to be useful, the VPL must be below a

certain threshold. It is for this reason that the purpose of this work is to make the error

bound as small as possible subject to Equation (2.5).

PROBLEM STATEMENT 25 2.5 CONSTRAINTS

In this section we briefly discuss the difficulties of solving the problem stated in the

previous section. We can divide these difficulties into two categories: constraints coming

from the coupling of the unpredictable physical behavior of the ionosphere with the

scattered and noisy measurements, and the operational constraints.

2.5.1 IONOSPHERIC CONSTRAINTS

The main difficulty in estimating each ionospheric delay and the error bound is the

unpredictable nature of the ionosphere coupled with the limited information that we have

about it. Because the ionospheric delays are correlated with distance – in a sense that will

be explained in Chapter 3 - it is possible to estimate the ionospheric delay experienced by a

user within the WAAS reference stations network. However, the characteristics of the

ionosphere change significantly over time. In particular, the stochastic properties of the

ionospheric delay – which are essential to form the error bound - are highly variable. We

therefore need to either check that an assumed model holds or estimate in real time the

magnitude of the decorrelation. This is all the more difficult due to the limited amount of

measurements, and their noisy nature, which makes the true ionosphere decorrelation less

observable. To make things even more difficult, there are sudden features in the

ionosphere that might be unobservable to the network of reference stations but that could

affect potential users. The ionospheric estimation algorithm needs to take all these threats

into account such that Equation (2.5) holds.

2.5.2 SYSTEM CONSTRAINTS

The constraints introduced in the previous subsection are unavoidable, because we do not

have any control over the ionospheric behavior. In this subsection, we introduce the

constraints inherent to the system. It is possible to design an algorithm disregarding the

limitations imposed by the current system, but such an algorithm could not be used in the

short term, and could only be used if its performance was worth the change. In this work,

PROBLEM STATEMENT 26 one of the main purposes of the algorithm is to fit within the frame that has been set up and

put to use in the current system.

The most important constraint is the messaging structure for the ionospheric information,

which is determined in the WAAS Minimum Operational Standard (WAAS MOPS). The

details of the standard can be found in [MOPS]. What is important here is that the

ionospheric information is sent as a grid of values on a thin shell, which will be described

in Chapter 3 -. Each value is nominally updated every 5 minutes. However, the user can

utilize a correction for up to 10 minutes, unless a warning is sent saying that the correction

is no longer valid. From this set of grid points, the user can compute an estimate of the

ionospheric delay for each of the satellites in view. In Chapter 5, we will see how the

ionospheric estimation algorithm was adapted to the WAAS MOPS.

Chapter 3

Ionospheric Delay Structure

In the previous chapter, we saw how GPS measurements are processed in real time and

post-processed to obtain clean GPS ionospheric measurements. This chapter makes use of

the large amount of post-processed data to determine a simple model that encompasses

both the deterministic and stochastic properties of the ionospheric delay. The methodology

applied here is primarily based on proven methods used in geostatistics. However, there

are several differences, the most important of which is the amount of data available. While

the geostatistician can look at all the data available (typically less than a hundred points),

we have to rely on mechanical analysis, due to the size of the data set. Also, the concern is

different: where the geostatistician looks for accuracy, we look for integrity. That is, we

want a conservative model of the ionosphere, in the sense that it leads to strong error

bounds in the estimation process. Equation Section 3

First, we need to explain a fundamental approximation that has simplified ionospheric

estimation: the thin shell model. This model transforms a three dimensional problem into a

two dimensional one, by transforming the slant ionospheric delay as measured by a GPS

receiver into a vertical ionospheric delay. The shortcomings associated with the thin shell

model will be briefly addressed as well. Second, we will show a series of maps of the

vertical ionospheric delay over CONUS. As with any exploratory analysis of spatial data,

we start with a close look at pictures of the phenomenon we are trying to characterize. This

IONOSPHERIC DELAY STRUCTURE 28analysis will allow us to determine qualitatively the driving features that can be

characterized deterministically. After that, we will turn our attention towards the statistical

properties of the vertical ionospheric delay. We will see that there is a simple and

conservative model of the ionospheric delay that captures most of the effects. This study

will continue by indicating the limits of the model, which frequently impacts any statistical

characterization. Finally, we will examine the temporal decorrelation of the vertical

ionospheric delay, and argue that, given our time scale, it is a second order effect.

3.1 THIN SHELL APPROXIMATION

The thin shell approximation states that all the free electrons lie within a thin shell at a

given altitude. Figure 3.1 illustrates this approximation: the slab represents the thickness of

the ionosphere and the blue traces of the ray paths show where they are affected by the free

electrons. The path integral corresponding to a GPS ionospheric measurement can be

approximated by [Klobuchar]:

( )User

eSatellite

2

nIono. delay= Vertical Iono delay*obliquity factor

l dl

f

α=

∫ (3.1)

Within this model, the path integral becomes a punctual measurement on the thin shell

labeled ‘vertical ionospheric delay,’ scaled by a multiplying factor labeled ‘obliquity

factor.’ The location of this point, which corresponds to the location where the ray path

crosses the thin shell, is known as Ionospheric Pierce Point (IPP). The obliquity factor

depends on the assumed altitude, H, and the elevation angle, α. It can be expressed as:

( )Obliquity factor = sec arcsin cose

e

RR H

α⎛ ⎞⎛ ⎞⎜ ⎟⎜⎜ ⎟+⎝ ⎠⎝ ⎠

⎟ (3.2)

IONOSPHERIC DELAY STRUCTURE 29Re is the radius of Earth. The altitude of the thin shell is set arbitrarily and is intended to

represent the mean height of the ionosphere. In the current WAAS broadcast message, the

altitude is fixed at 350 km [MOPS].

The drawbacks of this approximation are obvious: we ignore any vertical distribution of the

electron density. For example, two slant delays with the same IPP but different elevation

angles will have a different magnitude, because the region of the ionosphere crossed by

each ray path is different (except at the IPP). Yet, we retain the thin shell approximation.

First, it simplifies the estimation problem. Second, as we will see in Chapter 5, the current

ionospheric correction message relies on the thin shell model. A method relying on another

model would require changes in the broadcast standard - a standard already adopted by the

SBAS-capable receiver manufacturers [MOPS], [SARPS]. Such a change would only be

possible if an outstanding benefit could be expected. Finally, methods that relax the thin

shell approximation to take into account the vertical density profile of the ionosphere have

so far shown a low gain in performance compared to the thin shell model approximation

under the circumstances investigated [Lejeune].

Figure 3.1: Thin shell approximation over CONUS

Ionosphere

350 km

IPP

Equivalent Vertical Delay

IONOSPHERIC DELAY STRUCTURE 303.2 IONOSPHERE SNAPSHOTS

One of the useful properties of the thin shell model is to reduce each measurement to a data

point (the IPP) on a two-dimensional map. In this section, we show some snapshots of the

vertical ionospheric delay for different ionospheric conditions - quiet or stormy - and at

different times of the day. The distinction between quiet and stormy conditions is loose

here and there is a range of behaviors between them. For now it is sufficient to say that

quiet conditions generally correspond to low vertical ionospheric delays (below 10 meters)

and high correlation between neighboring measurements. For stormy conditions, it is the

opposite.

3.2.1 QUIET CONDITIONS

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4

4.5

5

5.5

6

6.5

Figure 3.2: Vertical ionospheric delay on July 2, 2000 at 5:00 p.m. E.T

IONOSPHERIC DELAY STRUCTURE 31Quiet conditions occur most of the time (more than 95%, even in the high solar season). In

Figures 3.2 and 3.3 we show snapshots of the ionosphere at two times on July 2, 2000. The

circles represent the post-processed IPP measurements at a given time, that is, all the slant

measurements from each of the reference stations to each satellite in view transformed to

an equivalent vertical ionospheric delay. The color code indicates the magnitude of the

vertical ionospheric delay measured at each location. The colored map is created by

interpolating the measured values. At this point, the only purpose of the underlying map is

to help us visualize the ionospheric delay.

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The most important feature of these maps is the presence of a clear trend. This trend is

only a part of the overall diurnal trend that we saw in Chapter 2, Section 2.1.1. Added to

this main planar trend, whose typical size is on the order of 4000 km, one can detect

IONOSPHERIC DELAY STRUCTURE 32smaller features which have a typical size of 500 to 1000 km. The shape of these features

is more difficult to describe than the planar trend, and could be considered random, but

correlated over distance. One can see that there is also a random noise above these

superposed trends. We can interpret the smaller features and the decorrelation as a random

spatial process correlated with distance.

3.2.2 DISTURBED CONDITIONS

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We now turn our attention to snapshots of the ionosphere during disturbed periods. Figures

3.4 and 3.5 show two typical vertical ionospheric maps during some of the worst

ionospheric behavior ever observed over the CONUS region. The vertical ionospheric

delay is now much higher than in quiet days and the decorrelation of the measurements is

larger. Still, we can characterize the vertical ionospheric delay by a combination of a

IONOSPHERIC DELAY STRUCTURE 33planar trend, and a stochastic (spatial) process that describes both the decorrelation above

the mean map and the smaller features, as we did for the quiet conditions. At this point, the

only difference is the magnitude of the decorrelation, and also of the secondary features.

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Figure 3.5: Vertical ionospheric delay on April 6, 2000 at 4:50 p.m. E.T

3.3 VERTICAL IONOSPHERIC DELAY MODEL

From the previous section, a possible model for the vertical ionospheric delay is:

( ) ( )0 1 2east northI x a a x a x r x= + + + (3.3)

IONOSPHERIC DELAY STRUCTURE 34I(x) is the vertical ionospheric delay at the IPP located in x = (xeast, xnorth), here these

coordinates can be considered to be respectively the longitude and the latitude. The

coefficients a0, a1 and a2 describe the planar trend. The scalar field, r(x), includes the small

features superposed on the planar trend and the decorrelation between neighboring

measurements. It is the behavior of this field, r(x), that is going to determine the estimation

algorithm, so we need to study its properties. The questions we would like to answer about

this field are: Is it correlated with distance? Is it reasonable to assume that it is Gaussian

(meaning that if we take n measurements and consider them random, they can be seen as

coming from a multivariate Gaussian distribution)? We can begin to answer these

questions by studying the correlation between pairs of residuals r(x).

3.3.1 SCATTER PLOTS

One of the tools used in geostatistics for exploratory data analysis is the scatter plot. To get

the scatter plot, we first compute at a given time frame (i.e. a set of measurements similar

to the ones plotted in Figures 4.2 through 4.5) all the pair-wise differences of residuals, and

the distances between the IPP measurements. We then plot the pairs formed by distance

and difference of residual on the x-axis and the y-axis, respectively. The cloud of points on

this diagram, called a scatter plot, will give us an idea of how the difference of residuals

increases as a function of distance. Notice that we want to compare the residuals, r(x), of

the IPP measurements and not the measurements themselves. If we compared the

measurements without removing the planar trend, we would observe that the residuals

increase with distance at a linear rate. This linear rate would be due to the deterministic

planar trend and would not say anything about the stochastic properties of the field of

residuals.

The difficulty now is to break any measurement into planar trend and residual field. De-

trending the data is a classical problem in spatial statistics [Cressie]. It is problematic to just

fit a plane to the data and compute the residuals. Some measurements might have high

leverage on the fit, either due to their large value or their geometry [Hastie], contaminating

the original data. For example, a measurement that represents an outlier to the plane will be

smoothed if we include it in the planar fit. In our description of the ionosphere, we want to

IONOSPHERIC DELAY STRUCTURE 35be aware of such outliers, and want to understand their spatial dependency. We therefore

need a method that does not contaminate the original data or that contaminates it in a

conservative way. The common practice in spatial statistics is to compare only the pairs of

measurements that lie in a direction unaffected by the trend [Journel], [Cressie]. This way

we ensure that the original data is not affected by the fit. For our purposes, there are two

problems with this approach. First, we end up with very few measurements, thus getting

lower statistical significance. Second, we assume that the variability in the direction of the

trend is the same as in the direction orthogonal to the trend. These two problems go against

our worst case approach.

Instead, we proceed as follows. For each pair of measurements we fit a plane to the

remaining measurements (up to a certain radius) and assume this planar fit describes the

trend. To our knowledge, our approach is new. The advantage of this approach is that the

data used to generate the pairs will not influence the trend. Also, we do not limit the

number of pairs. The inconvenience is that we do not take into account the uncertainty that

the plane might have due to the geometry of the measured locations, thus not defining the

trend correctly. However, this is likely to bias the differences of residuals in a conservative

way, because it will tend to give larger residuals. We mitigate this effect by considering

only the densest regions in terms of IPP measurements. Figure 3.6 illustrates the procedure

outlined above in a one-dimensional example.

i jx x−

( ) ( )i jr x r x−

Vertical Ionospheric Delay

Location

Figure 3.6: Estimating the differences of residuals

IONOSPHERIC DELAY STRUCTURE 36As we pointed out in the introduction to this chapter, we cannot visualize the scatter plots

corresponding to each time frame: there are too many. Instead, we group them by periods

of 24 hours. For quiet days, we can group all the scatter plots in a single plot. The large

amount of data points makes it necessary for us to replace the “clouds of points” by bins

whose color indicates the number of points lying in that region of the diagram. Figure 3.7

shows a scatter plot corresponding to July 2, 2000. The fit radius for the planar trend was

taken to be 2000 km (in section 3.2 we noticed that the planar trend has a characteristic size

of 4000 km). We made sure that the planar trend would be well defined by requiring a

minimum of 20 IPP measurements.

0 500 1000 1500 2000 25000

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1

1.5

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4.5

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resi

dual

diff

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ce in

met

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Num

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f Poi

nts

per P

ixel

100

101

102

103

Figure 3.7: Scatter plot of residuals corresponding to July 2, 2000

We can see that the difference between residuals tends to increase as the distance increases

between them. We remind the reader here that this decorrelation is not due to the

underlying planar trend, since we have removed it. Instead, it is the decorrelation of the

IONOSPHERIC DELAY STRUCTURE 37residual field, r(x) (a planar trend would have produced linearly increasing residuals).

There are two features that are worth mentioning here that will play an important role in the

estimation. First the concave shape of the residuals suggests that what we are seeing is a

random field [Webster]. Second, we notice that the decorrelation does not tend to zero as

the distance goes to zero. There are at least three causes to this effect. First, there might be

remaining biases in the measurements (see Chapter 2, Section 2.3). Second, there is the

‘slant to vertical error’ [Klobuchar]. This error is introduced by the thin shell model:

measurements having IPPs close together might have different look angles, in such a way

that when we consider the vertical profile of the electron density, the integrated TEC

appears discontinuous. Third, the decorrelation of the ionosphere might be so fast at the

origin that what is in reality continuous appears discontinuous. A scatter plot similar to

Figure 3.7 (not shown here) which only keeps the pairs of measurements coming from the

same station or same satellite, thus obviating the slant to vertical error, is still compatible

with a discontinuity at the origin. Further evidence of this phenomenon is confirmed by the

very fast temporal decorrelation of a single measurement [Datta]. The discontinuity or

continuity cannot be observed because there are very few samples at small distances. The

general characteristics of the scatter plots for other days remain the same, while the

magnitude of the decorrelation changes.

The scatter plots give a picture of the vertical ionospheric delay over a long period. Even

though most of the time the differences between residuals are small (red region), there are

cases in which they can become large. These outliers are critical for any estimation

algorithm with integrity. For each distance we can see that the differences between

residuals follow a certain distribution. In the next subsections we examine the nature of

this distribution and introduce the variogram which will help us reduce the information

contained in the scatter plots into a single function of the distance.

3.3.2 GAUSSIAN ASSUMPTION

We can measure the difference between the actual distribution of differential residuals at a

given distance and a Gaussian distribution. For this purpose, we will use the approach

taken by [Hansen00]. The idea is to choose a set of probabilities and compute the quantile

IONOSPHERIC DELAY STRUCTURE 38corresponding to each of them. For a given distance, if the distribution were Gaussian

there would be a specific ratio between the different quantiles. This ratio can be computed

from the Gaussian distribution. We can visualize how far the actual distribution is from the

Gaussian by normalizing each empirical quantile to the equivalent normal standard

deviation and plotting the result. If the normalized quantiles line up, we can say that the

distribution is close to Gaussian. The resulting plot is called sigma containment. Figure 3.8

shows the sigma containment plot corresponding to the data from July 2, 2000 for different

time periods.

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4 - 11 hrs

σ (68)σ (95%)σ (99.9%)

Figure 3.8: Sigma containment plot for July 2, 2000

We observe that the Gaussian assumption is appropriate here. There are cases where the

tails of the experimental distribution exceed what would be expected. However, most of the

time, the Gaussian hypothesis is a conservative one. The Gaussian distribution allows for

infinite tails, when we know that the tails of the true distribution must be clipped.

IONOSPHERIC DELAY STRUCTURE 393.3.3 THEORETICAL AND EXPERIMENTAL VARIOGRAM

The variogram [Webster] is defined as the expectation of the squared difference between

two values of a random field at different locations as a function of the distance. This

expectation is divided by two:

( ) ( ) ( ) ( )( )( )21 2 1 2 1 2

1,2

x x x x E r x r xγ γ= − = −

The variogram measured from data is called an experimental variogram. In our case, we

are interested in the de-trended values (the field of residuals, r(x)). The classical formula

defining the experimental variogram is:

( ) ( ) ( )( )2exp 1 2

, | ,2 2

12

i j i jx x x x d d

d r xδ δ

γ⎡ ⎤− ∈ − +⎢ ⎥⎣ ⎦

= −∑ r x

Because we do not have pairs of measurements separated by exactly d, we need to bin the

pairs with a certain tolerance δ. δ was chosen to be 50 km. This is justified by the small

change in the residuals over such a distance (see Figure 3.7). This expression supposes that

the field has a uniform mean and that there is no dependency on the orientation of the line

joining the two measurements, i.e. that the field is isotropic. This is not strictly true at a

given time as the variability tends to be larger along a certain axis. However, this axis

changes in an unpredictable way, such that it is easier and simpler to assume that the field

is isotropic and average over axis orientation. The link between the scatter plot introduced

in the previous subsection and the variogram is the following: to obtain the variogram we

take the variance of the distribution of differences of residuals for each distance and we

divide the result by two. Figure 3.9 shows variograms corresponding to the scatter plot

shown in Figure 3.7 at different times. One of the problems of this definition of the

variogram is that the variance of the distribution is not sensitive enough to the outliers.

However, we have seen in the previous subsection that for each distance, the Gaussian

assumption was an acceptable one, in which case the variance is enough to characterize the

distribution.

IONOSPHERIC DELAY STRUCTURE 40

0 500 1000 1500 2000 25000

0.05

0.1

distance in km

vario

gram

in s

quar

e m

eter

s

Figure 3.9: Variograms of the residual vertical ionospheric delay on July 2, 2000

As the scatter plot suggested, we see that the intercept at zero distance is non-zero. This is

called the ‘nugget effect’ in geostatistics [Webster]. By extension, the magnitude of the

intercept is called the nugget (this name comes from gold mining, where the gold grade of

the soil is discontinuous due to the existence of gold nuggets). We also see that the

variogram increases with distance. This means that the decorrelation of the ionosphere

increases as the distance between two points increases, even after we remove the planar

trend. It is also reassuring to see that the variogram does not increase quadratically. Such a

variogram would indicate that there is a deterministic trend unaccounted for, and that the

field can no longer be considered random [Webster].

IONOSPHERIC DELAY STRUCTURE 41The variogram gives us a quantitative measure of the spatial decorrelation. Because the

field is nearly Gaussian or can be overbounded by a Gaussian, the knowledge of the

variogram conservatively characterizes the random field.

3.3.4 MODEL VARIOGRAM AND COVARIANCE

Thus far, we have shown how to compute the experimental variogram of the residual field

r(x). If we just wanted to have an idea of the behavior of the residual field, we could stop

here. However, as stated earlier, the purpose of this chapter is to derive a simple model of

the vertical ionospheric delay that can be used for estimation. In particular, the model

variogram needs to be ‘admissible’ [Webster]. An admissible variogram is one that cannot

give rise to negative variances when used in variance calculations. In order to understand

better why there are constraints on the variogram, we translate the variogram in terms of

covariance. We have the following relationship between the two:

( ) ( )21 2 1 2, ,Cov x x x xσ γ∞= −

Notice that when we write this formula we need to assume that the variogram has a sill, 2σ∞ . In other words it is assumed that the variogram tends toward a finite value 2σ∞ as the

distance approaches infinity. This is not a problem, since we know that the vertical

ionospheric delay is bounded, even if the sill cannot be seen in the experimental variogram

shown in Figure 3.8. As long as the sill appears at large distances (larger than 5000 km),

this is not a critical choice when modeling the variogram. The value of the sill is the

variance of two measurements that are completely decorrelated.

Now that we have the covariance between two residuals, we can compute the covariance

matrix of a set of residuals, r(x1),…, r(xn):

( ) ( )( ) ( ) ( )( ) ( )0,i j i j iCov r x r x E r x r x C x x= = j− (3.4)

The only requirement for the covariance to be admissible is that any linear combination of

the residuals must have a positive variance. This is equivalent to requiring that the

IONOSPHERIC DELAY STRUCTURE 42covariance matrix be positive definite. An admissible and widespread model for the

variogram is the exponential model [Webster]. This model variogram has the three

properties we need. There is a nugget effect at the origin, the variogram increases linearly

near the origin, and there is a sill. The expression for the model variogram is:

( ) 1dad c eγ ν

−⎛ ⎞= − +⎜ ⎟

⎝ ⎠ (3.5)

The value of the variogram between two points is only a function of the distance between

the two points, d. We should point out again that the value of the variogram at the origin is

zero, and that there is a discontinuity, the nugget. The different parameters of the

variogram were chosen visually, so that the function given by (3.5) would approximately

overbound different experimental variograms for a quiet day. The parameters retained for

CONUS were: c = 2 m2, a = 32000 km, and ν = .04 m2. Figure 3.10 shows the

experimental variogram (blue) and the model variogram chosen (black). In Chapter 5, we

will primarily use the covariance, given by:

( ) ( )daC d c d ceν γ

−= + − = (3.6)


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vario

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in s

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Figure 3.10: Experimental variogram and model variogram

3.3.5 DECORRELATION DURING A STORMY DAY

In the last two subsections we have seen the scatter plot and the variogram for a quiet

ionospheric day (July 2, 2000). We now turn our attention briefly to more disturbed days:

April 6, 2000 and September 8, 2002.

We show in Figure 3.11 sigma containment plots corresponding to the onset of the April 6,

2000 ionospheric storm, which is one of the worst ionospheric storms that the CONUS

region has seen. We can see that the differences between residuals increase more rapidly

with distance, and, what is more important, that the discontinuity at zero distance is much

larger. Again this discontinuity might be due to the slant to vertical error due to the thin

shell model, but it is also due to a very fast decorrelation of the electron density during

IONOSPHERIC DELAY STRUCTURE 44storms. We can also see that in the third plot, the Gaussian hypothesis is more problematic.

This due to the mixing between variograms corresponding to different behaviors.

0 500 1000 1500 2000 25000

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15 - 18 hrssi

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6

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10

Distance (km)

20 - 24 hrs

σ (68)σ (95%)σ (99.9%)

Figure 3.11: Sigma containment plots during April 6,2000

Overall, the sigma containment corresponding to the quiet day (Figure 3.8) and the sigma

contatinment corresponding to the stormy day (Figure 3.12) are similar in shape. That

suggests that we can describe the ionosphere with a similar model (planar trend plus

residual field) in both cases. The only thing that changes is the value of the parameters

describing the residual random field, i.e., the variogram.

Figure 3.12 shows the sigma containment for September 8, 2002 where the shape of the

decorrelation is the same as for a quiet day, and the Gaussian hypothesis holds.


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σ (68)σ (95%)σ (99.9%)

Figure 3.12: Sigma containment plots during September 8, 2002

These remarks lead us to postulate that there exists a parameter u such that the variogram

can be written:

( ) ( )20d u dγ γ= (3.7)

The function γ0 is the variogram corresponding to quiet conditions, which was introduced

in the previous section. The parameter u describes the degree of disturbance in the

ionosphere: a larger u corresponds to a disturbed ionosphere, and u=1 corresponds to a

quiet ionosphere. This formula supposes that the slope of the variogram is proportional to

the nugget effect, which is close to what we observe.

IONOSPHERIC DELAY STRUCTURE 463.4 IONOSPHERIC IRREGULARITIES

We now have a simple model of the vertical ionospheric delay which has two components:

a planar trend (the deterministic component) and a random Gaussian residual field – the

stochastic component. This last component is characterized by a variogram (or,

equivalently, by the covariance) indexed by the parameter u, which describes the roughness

of the vertical ionospheric delay. It is important at this point to be aware of the limitations

of this model. The main limitation is the occasional lack of stationarity in the random field.

The vertical ionospheric delay might be well described by a different variogram in different

regions of the ionosphere. For a region like CONUS, what we would like to know is the

shape of such irregularities: Can we have an irregularity in the middle of an otherwise quiet

region? Can we have an irregularity coming from the side of an otherwise quiet region?

Notice that here irregularity is any behavior that is not accounted for in the assumed model.

It is exceedingly difficult to tell simply by looking at a certain situation whether it is

compatible with a certain random model or not. The ultimate answer to that question will

be given in Chapter 6, when we will test the bounding algorithm based on the model given

above. In this section we will mostly hint at the possible drawbacks of such a

characterization, and cite previous research that has attempted to characterize the lack of

stationarity in the vertical ionospheric delay maps.

3.4.1 ISOLATED IRREGULARITIES

First we examine the threat constituted by isolated irregularities. Isolated irregularities

include regions of the ionosphere where the surrounding region is well observed and well

described by our model, but contained inside is a region that is not. Let us imagine that we

have a dense set of measurements over CONUS and that a certain variogram similar well

describes the random field of residuals. Suppose now that in the middle of it, there is an

unobserved region (for example a disk of a certain radius) without any measurements. Is it

possible that the residuals corresponding to that region might not be compatible with the

variogram valid elsewhere? Again this question is difficult to answer, mainly because a

given variogram does not prohibit a given situation from happening, it only makes it less

likely. An attempt to answer this question was made in [ITM], and we now summarize the

IONOSPHERIC DELAY STRUCTURE 47methodology and conclusions of that work. The approach taken in that work used data

deprivation in order to simulate the lack of data. The following procedure was applied: for

any given measurement, all the measurements surrounding it at the same time frame up to a

certain radius were excluded. The purpose of this step was to simulate the lack of data in

that region. Then, the remaining measurements were checked to determine compatibility

with a planar trend with a certain decorrelation around it (this was done by computing the

sum of squared residuals). If they were found to be compatible, the residuals of the

excluded measurements were computed and their size was compared to the observed

residuals. This analysis determined that isolated irregularities were not a serious threat.

3.4.2 GRADIENTS

The other threats we need to consider are the ‘gradients.’ In this context, gradients are a

sharp change in vertical ionospheric delay. If this sharp change occurs in an already

disturbed region, it might not be a problem because the measurements will reflect the

general disturbance. If, on the other hand, the gradient divides the region into a disturbed

side and a quiet side, or two distinct quiet regions, there might a problem. The same

analysis that was done for the isolated irregularities was carried out for gradients in [ITM].

Now, instead of excluding disks around every single measurement, whole half planes were

excluded; then the same procedure (computing the observed residuals and comparing them

with the ‘unobserved’ ones) was applied. This research determined that gradients could

constitute a serious threat to stationarity. In Figure 3.13 we show the vertical ionospheric

delay map toward the end of the ionospheric storm of July 15-16, 2000. One can see that

the region above 35 degrees latitude corresponds to a quiet day ionosphere. South of it we

see that the vertical ionospheric delays increase to more than 20 meters, and so does the

variability. If we were to estimate the vertical ionospheric delay with the IPPs with a delay

below 8 meters, the estimation error could be over 15 meters in Florida, which could lead

to slant errors above 50 meters. (This picture also illustrates the difficulty of such studies

as we are always limited by the sampling that was done that day.) As we will see later,

situations like the one pictured in Figure 3.13 are currently handled in WAAS by the threat

model [Sparks01], which will be outlined in Chapter 4, in conjunction with the algorithm

designed in this thesis.


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Figure 3.13: Very large gradient during the July 15-16, 2000 ionospheric storm

3.5 TEMPORAL DECORRELATION

So far, we have dealt with the problem of determining the structure of the vertical

ionospheric delay as a static problem. We have ignored the fact that the vertical

ionospheric delays evolve with time (at a fixed location in the thin shell). A model for the

random behavior of the ionosphere that includes both the spatial decorrelation and the

temporal decorrelation would be difficult to obtain and express for any time period. Among

other problems, we would have to include the deterministic temporal behavior in our

description. However, as we saw in Chapter 2, the ionospheric corrections and bounds

need only be valid for ten minutes. Based on the speed of the reference in which the

ionosphere appears stationary, 360 degrees of longitude per day, we see that the underlying

trend shifts by 2.5 degrees longitude, or about 200 km at typical CONUS latitudes, over

IONOSPHERIC DELAY STRUCTURE 49ten minutes. This is a small distance compared to the characteristic distances of the

problem (see Figures 3.7 and 3.8). The vertical ionospheric delay maps shown above

suggest that as a first approximation we can consider that the underlying trend is fixed, and

that the evolution is not deterministic but stochastic.

To visualize the short term temporal and spatial structure of the vertical ionospheric delay,

we form a generalized scatter plot. That is, we add the time as a new dimension with the

pairs of IPP measurements separated by distance and time. Instead of having two axes like

in Figure 3.7, we will have three: difference of residuals, distance between IPP locations

and time delay between the two IPP measurements. Practically, we examine 10 minute

periods (600 seconds) where each time step is 200 seconds. Each period contains three

time frames. For a given measurement we fit a plane (planar fit radius is taken to be 2000

km) at the time frame and remove this from all three periods. This step should

approximately remove the trend. After that, we form the difference of residuals between

our test measurement and each of the other ones. We apply this to all the measurements

and plot each difference of residual in a three dimensional scatter plot. Since the resolution

in the time domain is only four, we can plot it as a series of four scatter plots similar to

Figure 3.7. In Figure 3.14, we show the sigma containment plots corresponding to these

scatter plots. We can see that there is not much difference between the four plots. For time

lags below 10 minutes, there is no noticeable change in the structure of the residual random

field. These plots suggest that the model chosen to describe the vertical ionospheric delay

is stationary over periods of 10 minutes, which simplifies greatly the design of the

algorithm, since now the problem becomes purely spatial and not temporal.


0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

Distance (km)

sigm

a co

ntai

nmen

t in

met

ers

τ= 0 s

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

Distance (km)

τ= 200 s

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

Distance (km)

sigm

a co

ntai

nmen

t in

met

ers

τ= 400 s

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

Distance (km)

τ= 600 s

σ (68)σ (95%)σ (99.9%)

Figure 3.14: Spatial and temporal decorrelation during July 2, 2000

Unfortunately, a difficulty arises due to the lack of stationarity during the onset of storms in

the time domain. During these times the vertical ionospheric delay can increase rapidly.

Here again, as for the spatial irregularities, this difficulty is treated by a piece of the

algorithm that deals specifically with threats that lie outside the assumed model. This part

of the algorithm will be dealt with in Chapter 5 and Chapter 6.

3.6 CONCLUSION

In this chapter, we have extracted from the post-processed TEC data a simple

characterization of the vertical ionospheric delay on a quiet day:


( ) ( ) ( ) ( )0 1 2east northI x a a x a x r x= + + + (3.8)

In this equation, r(x) is a random residual field that has zero mean and a covariance

between two measurements, i and j:

( ) ( )( ) ( ) ( )( ) ( )0,i j i j iCov r x r x E r x r x C x x= = j− (3.9)

From now on we will call this model the nominal covariance: it is a covariance that

describes conservatively a quiet vertical ionospheric delay. We have also seen that this

characterization is also sound for a stormy day provided we suppose that the covariance is:

( ) ( )( ) ( )20,i j iCov r x r x u C x x= − j (3.10)

The parameter u describes the roughness of the vertical ionospheric delay: the larger it is

the more disturbed the ionosphere is.

We have also seen that within a certain time lag, the vertical ionospheric delay field is

stationary and that there is no need for a complex spatial and temporal stochastic model

(the spatial one is enough).

The last part of this chapter has examined where this model could fail. Potential failures

occur when there are severe stationarity violations, both spatial and temporal. However, it

has been determined elsewhere [ITM], [Sparks01] that those seem to be limited to

irregularities coming from the side of the well observed regions (an irregularity cannot

appear in the middle of CONUS). The ultimate answer to the question to whether the

assumed model is conservative enough will come when we check the final algorithm

against real data in Chapter 6.

Chapter 4

Ionospheric Estimation Algorithm

In Chapter 3 we found a characterization of the vertical ionospheric delay that is comprised

of a deterministic part (the planar trend) and a stochastic part (the residual field.) Here, we

are going to use this model to design the estimation algorithm. The estimation algorithm

must provide at each time epoch and at each location an estimate of the vertical ionospheric

delay and, more importantly, a hard bound on the possible error of this correction. As seen

in Chapter 2, this bound must provide a Probability of Hazardously Misleading Information

below 2.25X10-8. This error bound must be as small as achievable, such that the

availability is as high as possible given the current information. The estimation problem

can be seen as a constrained optimization problem: for each location in the thin shell we

minimize the error bound such that the PHMI is below 2.25X10-8. For a given location, we

can write the following problem:Equation Section 4

Minimize ∆ such that ( ) 8Prob 2.25*10true estI I −− > ∆ ≤ (4.1)

In this equation, Iest is the estimated vertical ionospheric delay, Itrue is the true vertical

ionospheric delay, and ∆ is the estimated error bound. This equation must hold for any user

of the estimate. There are several difficulties to overcome. First, the underlying

ionospheric model is uncertain: the covariance model is indexed by the parameter u, which

is unknown. This means that the roughness of the vertical ionospheric delay will have to

IONOSPHERIC ESTIMATION ALGORITHM 53be safely bounded. This is all the more difficult as the vertical ionospheric delay

measurements are randomly scattered over the region of interest. Second, the real time

measurements used by the estimation algorithm have measurement noise (see Chapter 2.)

This would not be a problem by itself if the underlying ionospheric behavior were known.

We have a safe upper bound on the measurement noise, and it is easy to translate this to the

error bound of the estimate. What is more problematic about the measurement noise is the

fact that it blinds our observability of the underlying ionospheric behavior (the parameter

u).

Due to the multiple layers of complexity of the estimation algorithm, in this chapter we will

proceed as follows. In the first section we will assume that there is no model uncertainty

(the parameter u is known). This situation is easily modeled and we can find the exact

solution, which turns out to be kriging. In the second section we will drop the assumption

that we know the underlying model and introduce a factor that accounts for this

uncertainty, the inflation factor.

4.1 ALGORITHM ASSUMING KNOWLEDGE OF THE MODEL

We remind the reader here that when we say that we assume the model to be known we

mean that the covariance of the residual field is known:

( ) ( ) ( )( ) ( ), ,k l k l k lCov x x E r x r x C x x= = (4.2)

From now on we will call the residual field the process noise, as opposed to measurement

noise. The problem is as follows: Given n IPP measurements, I(x1)…I(xn), what is the best

estimate and the smallest error bound on the estimate that can be given at a certain location

x? In this section we will see that we can solve Problem (4.1) exactly. We will start with

the general form of the estimator. After that, we will express the estimation variance as a

function of the variables of the problem. The last step will consist of a straightforward

constrained minimization problem that will lead to the kriging equations. The beginning of

IONOSPHERIC ESTIMATION ALGORITHM 54this section is not new. However, some modifications are needed that are not easily found

in the literature. In addition, the final expression of the kriging equations is original. This

section will also allow us to introduce several notations.

4.1.1 UNBIASED LINEAR ESTIMATOR

The estimate at location x is going to be a function of the surrounding measurements

Imeas(x1)…Imeas(xn). We are going to limit the search of such a function among linear

functions, that is, the estimate is going to be a linear combination of the measurements:

( ) ( )1

n

est k meas kk

I x Iλ=

=∑ x (4.3)

We only need to find the coefficients λ1,…,λn. Now we remind the reader of the different

components of the measurement. We have:

( ) ( ) ( ) ( ) ( )0 1 2east north

meas k k k k kI x a a x a x r x m x= + + + + (4.4)

The first four terms were examined in Chapter 3. The first three terms define the

underlying planar trend (deterministic component) and the fourth term represents the

residual field (the process noise.) The fifth term, m(xk), is the measurement noise specific

to the receiver and the satellite pair (this term does not depend on ionospheric behavior).

As stated in Chapter 2, this noise can be assumed to be zero mean and Gaussian, provided

that a large enough standard deviation is taken to bound the tails of the true error

distribution. The estimator should be unbiased, i.e.:

( )( ) ( ) ( )( ) ( ) ( ) ( )( )0 1 21

neast north

est k meas k truek

E I x E I x E I x E a a x a x r xλ=

⎛ ⎞= = = + +⎜ ⎟

⎝ ⎠∑ +

X⎤ =⎦

(4.5)

By noticing that E(r(x))=E(r(xk))=E(m(xk))=0 we can write, after some algebra, that:

( ) ( )1Teast northTG x xλ ⎡= ⎣ (4.6)

IONOSPHERIC ESTIMATION ALGORITHM 55The definitions of G and λ are the following:

( ) ( )

( ) ( )1

1

1 1east eastT

n

north northn

G x x

x x

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

, [ ]1T

nλ λ λ=

The constraints in (4.5) show that the λ coefficients must be compatible with the planar

trend. Notice that one could generalize this result to any kind of trend by augmenting the

matrix G appropriately.

4.1.2 ESTIMATION VARIANCE

The estimation variance is defined as the following expectation:

( ) ( )( )( )22est true estE I x I xσ = − (4.7)

The fact that the estimator is unbiased allows us to write:

( ) ( ) ( ) ( ) ( )( ){ }22

1 1est n nE r x r x r x m x m xσ λ⎛ ⎞⎡ ⎤ ⎡= − +⎜ ⎟⎣ ⎦ ⎣⎝ ⎠⎤⎦

This expression can be developed. Noticing that the measurement noise, m, and the

ionospheric process noise, r, are uncorrelated, after some algebra we have:

( ) ( )( ) ( ) (2 , , 2 ,T Test k l k l kC x x M x x C x x C x xσ λ λ λ= + − + ), (4.8)

Here, C(xk, xl) is an n by n matrix whose elements are computed using the assumed

covariance for the process noise. Similarly, M(xk, xl) is the covariance matrix of the

measurement noise, m. C(x, xk) is a vector whose elements are the covariance between r(x)

and each of the residuals at the measurement locations, r(xk). C(x, x) is the value of the

covariance at a distance of zero and does not depend on the location x.

IONOSPHERIC ESTIMATION ALGORITHM 564.1.3 ERROR BOUND AND ESTIMATION VARIANCE

In this subsection we explain the relationship between the estimation variance and the error

bound. First, recall the assumption that the process noise is Gaussian. Under this

assumption the random variable ( ) ( )true estI x I x− is zero mean Gaussian with a standard

deviation σest. Considering all the possible realizations of the random field, we have:

( )Prob 2true estest

I I Qσ⎛ ⎞∆

− > ∆ = ⎜⎝ ⎠

⎟ (4.9)

Here Q is the cumulative distribution function of a Gaussian. If we want this probability to

be 2.25X10-8, we need to have ∆ = Kσest where K=5.92. Minimizing ∆ is then equivalent

to minimizing σest2. This is what is done in the next subsection.

4.1.4 FINDING THE OPTIMAL COEFFICIENTS: KRIGING EQUATIONS

The problem of determining the λ coefficients can be cast as a constrained optimization

problem:

Minimize ( ) ( ) ( )( ) ( ) (2 , , 2 ,T Test k l k l kC x x M x x C x x C x xσ λ λ λ λ= + − + ),

X

Subject to (4.10) TG λ =

This is a classical optimization problem and there are many methods to solve it. We show

here the one using Lagrange multipliers. The Lagrangian is:

( ) ( ) ( )2, T TestL Gλ µ σ λ µ λ= + − X

We find the solution of the minimization problem by setting the partial derivatives of L to

zero. We skip the details of this calculation. At the end, we get the following matrix

equation for the coefficients:

IONOSPHERIC ESTIMATION ALGORITHM 57

( ) ( ) ( ), , ,0

k l k l kT

C x x M x x G C x xG X

λµ

⎡ + ⎤ ⎡⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

⎤⎢ ⎥⎣ ⎦

)

The solution is then:

( ) ( ) (1, , ,

0k l k l k

T

C x x M x x G C x xG X

λµ

−⎡ + ⎤ ⎡⎡ ⎤

=⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(4.11)

This equation defines the coefficients, but does not give an idea of the behavior of the

coefficients. We would like to know how the λ coefficients depend on the covariance

properties of the process noise and the measurement noise. For that, we invert the matrix

by blocks:

( ) ( ) ( ) (( ) ( )

1 11

1 1

, ,0

T T T

k l k lT T T T

W WG G WG G W WG G WGC x x M x x GG G WG G W G WG

− −−

− −

)⎡ ⎤−⎡ + ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦ −⎢ ⎥⎣ ⎦

(4.12)

The definition of W is:

( ) ( )( ) 1, ,k l k lW C x x M x x

−= +

The λ coefficients are defined by:

( )( ) ( ) ( )1 1,T T T

kW WG G WG G W C x x WG G WG Xλ− −

= − + (4.13)

This expression shows the different components of the estimate: the stochastic component

in the first term and the deterministic component in the second one. Since the matrices

involved in this formula are going to appear several times in the next pages, we include the

notation:

( )( ) ( )1 1 , T T TP W WG G WG G W H WG G WG

− −= − =

By replacing these expressions in the formula for the estimation variance (4.8) we get:


( ) ( ) ( ) ( ) ( )12 , , , 2T TT Test k k kC x x X G WG X C x x PC x x C x x HXσ

−= + − − , (4.14)

We can give an approximate interpretation to this formula. The first two terms correspond

to the estimation variance that we would get if we were to estimate the vertical ionospheric

delay by the value of the estimated trend. The last two terms correspond to the reduction

we can achieve by taking into account the spatial properties of the process noise –or

residual field. For example, if the process noise was completely decorrelated from one

location to another, then C(x,xk) would be zero, so there would not be any reduction in the

estimation variance, which is what we expect.

The formulas presented in this subsection differ from the classical kriging equations

[Webster], [Cressie], [Journel] in two ways. First, we have included the effect of

measurement noise (it has not previously been included in kriging) which has a different

role in the estimation formulas. Second, classical formulations of kriging stop at Equation

(4.11), whereas here we have obtained a decomposition of the different contributions in the

coefficients.

4.1.5 KRIGING MAPS

In this subsection, we give an example of the estimation variance map produced by the

equations derived in the previous section over the CONUS region. We will also provide

some details, among which are the choice of measurements and the coordinate and metric

system used on the thin shell, which are the same that are used in the current WAAS

ionospheric estimation algorithm [Walter00].

The distance used between IPP measurements on the thin shell is the true distance between

the two locations, as opposed to the great circle distance. The coordinate system used is a

local one. For each IPP location we use the East –North –Up coordinate system [Enge01].

The curved surface of the thin shell is flattened by just considering the East and North

coordinates. The map gets ‘deformed’ slightly as we depart from the location of interest,

but it has been shown to be insignificant. As the shape of the variogram suggests in Figure

3.8, a difference in distance of 100 km is not going to modify the covariance matrix

IONOSPHERIC ESTIMATION ALGORITHM 59significantly. For each location we do not use the whole set of IPP measurements. Instead,

the measurements lying within a certain radius (the search radius) are selected. The

determination of the search radius is the same as in the current WAAS estimation

algorithm. If there are more than 30 measurements lying within 2100 km, it is the smallest

radius containing 30 measurements (but larger than 800 km). Otherwise, it is 2100. An

estimate is only given if there are at least 10 measurements. At this state of the design, the

search radius is not a key parameter: the coefficients assigned to measurements lying at

more than 1000 km have virtually no weight on the estimate.

conf

iden

ce b

ound

in m

eter

s0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

-140 -120 -100 -80 -6015

20

25

30

35

40

45

50

55

Longitude in deg

Latit

ude

in d

eg

Figure 4.1: Map of the estimation variance on July 2, 2000 at 12:00 pm E.T.

To generate the map shown in Figure 4.1, we took all the measurements at a certain time

during July 2, 2000 from the post-processed data. The covariance model chosen here is the

one specified in Section 4.3.4, Equation (4.5). For this example, the measurement noise is

taken to be zero. The covariance used is the nominal covariance derived in Chapter 3.

IONOSPHERIC ESTIMATION ALGORITHM 60This is approximately true for the post-processed data and it allows us to see the effect of

the spatial decorrelation on the error bounds.

In this map of σest we see how the regions rich in measurements (the white stars) have the

lowest estimation error, and how the estimation error increases as we depart from them.

The degradation in the error bound is a function of the variogram derived in Chapter 3. It

is the behavior of the estimation variance and its link to the estimated stochastic properties

of the ionosphere that makes kriging so attractive for an ionosphere estimation algorithm

requiring tight error bounds. If the vertical ionospheric delay process noise were always

bounded by the model covariance (which it is in all quiet days, that is, 99% of the time), we

could stop the analysis here.

Unfortunately, as we saw earlier, the process noise varies greatly. To solve this problem,

we could assume a nominal process noise that bounds all the ionospheric behavior ever

seen, but given the characteristics of the worst ionospheric conditions, that would mean low

availability [Walter00]. We therefore need to look for another solution, one that makes a

better use of the information when we compute the correction.

4.1.6 ESTIMATION VARIANCE DEPENDENCE ON MEASUREMENT NOISE

In this subsection we study how the estimation variance computed using kriging depends

on the measurement noise and on the process noise. The purpose here is to show that,

when the process noise is well known, the measurement noise has very little effect on the

estimation variance. The results from this section will support approximations made

below. First, let us divide the estimation variance into two terms: the part coming from the

process noise and the part coming from the measurement noise. We have:

( ) ( )2 2 2est process measσ σ λ σ= + λ

In this equation:

( ) ( ) ( ) ( )2 , 2 , ,T Tprocess k l kC x x C x x C x xσ λ λ λ λ= − +


( ) ( )2 ,Tmeas k lM x xσ λ λ= λ

There is an important qualitative difference in these two expressions. The first one cannot

go to zero as the number of measurements increases, whereas the second one can. This

comes from the fact that measurement noise is averaged out as the number of

measurements increases. On the other hand, the process noise cannot go to zero because

even with an infinite number of measurements there will always be random behavior in

between the measurements. The first term has a lower bound in ν, the nugget effect in the

covariance. The purpose here is to show that in the real system there is a large

dissymmetry between the two terms. The first one is always much more important than the

second one. Figure 4.2 shows the ratio ( ) ( )2 2. /meas processσ λ σ λ computed using the nominal

model on July 2, 2000. Notice that this ratio is only a function of the measurement

covariance, the process noise covariance and the geometry of the measurements.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

200

400

600

800

1000

1200

1400

1600

Ratio between measurement noise and process noise

Figure 4.2: Histogram of the ratio σmeas2/ σprocess

2 on July 2, 2000

IONOSPHERIC ESTIMATION ALGORITHM 624.2 IONOSPHERIC MODEL INFERENCE

In this section we drop the assumption that the process noise is known. Now we assume

that the process noise is the one given by Equation (3.10). The parameter u is now

unknown. At each time step it is possible to get an estimate of u; however, there is an

uncertainty associated with u, which is dependent on the number of measurements and their

quality (the measurement noise.) The idea of this section is largely based on the current

WAAS ionospheric estimation algorithm [Walter00], and some proposed changes

[Cormier]. New here is the method to evaluate the Probability of Hazardously

Misdetection Information and the more realistic treatment of the effect of measurement

noise.

In this section we will again take an incremental approach, by first solving an easier

problem. The first subsection will show the underlying idea through a simple example.

The second one describes how to ‘reduce’ the measurements to a more usable form. After

that, we will solve the problem assuming that there is no measurement noise (M=0). The

following subsections introduce the necessary changes when the measurement noise is no

longer considered zero.

4.2.1 MOTIVATION

Let us assume that M=0, i.e., there is no uncertainty due to imperfect measurements: any

deviation from the trend is due to the process noise. The idea is to estimate the value of u

and then use an inflated model which is a function of this estimated value. Instead of using

the nominal covariance matrix, C0, we use an inflated model, u02C0, where u0 is based on

the measurements. The motivation behind this is as follows: Let us suppose that we have a

set of m independent zero mean Gaussian random variables, Yk, with standard deviation

uσdecorr. We have:

2 2 2

1

1 m

k dk

E y um

σ=

⎛ ⎞=⎜ ⎟

⎝ ⎠∑ ecorr (4.15)

IONOSPHERIC ESTIMATION ALGORITHM 63As a result, a good estimate of u is therefore the experimental variance of the sample. It is

therefore natural to use it in the inflation factor. However, there is an uncertainty

associated with this estimate: the experimental variance is itself a random variable which,

normalized by u2σdecorr2, is chi-square distributed [Ross]. It is this fact that gives us a

handle on the unknown parameter u.

4.2.2 DE-TRENDING AND DECORRELATING THE MEASUREMENTS

In our case, we need to reduce the problem to the previous one. First, we do not measure

directly the field r(x), but r(x)+m(x) plus a trend, as shown in Equation (4.4). Second, the

residuals are Gaussian but not uncorrelated; their covariance is Ctot=C+M. We therefore

need to first filter the trend, and decorrelate the resultant residuals. The formation of a chi-

square distributed statistic from correlated measurements with a trend is a classical one

[Hastie02]. However, as later in the analysis we will need the reduced residuals (de-

trended and decorrelated), we go over the derivation. In other words, we want to find Г

such that y= ГImeas is a random Gaussian vector whose covariance is given by the identity

matrix. We now show how to obtain the matrix Г.

First of all we want to filter out the trend. This is equivalent to requiring for any 3

by 1 vector a. This implies:

0GaΓ =

0GΓ = (4.16)

Since G is an n by 3 matrix, Г is at most of rank n-3, which means that we will only be able

to extract n-3 orthogonal residuals. So from now on we assume that y is a random Gaussian

vector with n-3 components. Condition (4.16) means that if a matrix F has null space G,

then there exists a matrix H such that Г=HF. A suitable matrix F is:

( ) 1T TF I G G WG G W−

= −

F is a projection matrix of rank n-3 whose null space is the range of G. Now, to find H we

set the covariance of y= ГImeas to be the identity, i.e.:


( )( ) 3T T

meas meas nE I I C I −Γ Γ = Γ Γ =

The equation for H is:

( )( )113

T T TnH W G G WG G H I

−−−− =

We now do the change of variable12H HW

−= . The equation is now:

( )1 1

12 2

3T T T

nH I W G G WG G W H I−

−

⎛ ⎞− =⎜

⎝ ⎠⎟ (4.17)

The matrix ( )1 1

12 T T 2I W G G WG G W

−− is an orthogonal projection. There exists U

orthogonal such that:

3 00 0nT I

U PU −⎡ ⎤= ⎢ ⎥⎣ ⎦

If we call U the first n-3 columns of U, then fulfills Condition (4.17). We take Г

to be:

TH U=

( )1 1

12 2T TU W W G G WG G W

−⎛ ⎞Γ = −⎜ ⎟

⎝ ⎠T (4.18)

Now, to obtain a chi-squared distributed statistic, we form:

( )( )1T T T T T Tmeas meas meas measy y I I I W WG G WG G W I

−= Γ Γ = −

This formula coincides with the classical formula [Hastie2]. This formula assumes that W,

the inverse of the true covariance, is known, which it is not. There are still advantages to

this transformation. Even if Г is based on a different covariance, y is still zero mean. Also,

in the case where there is no measurement noise, the covariance of y is still diagonal. In the

IONOSPHERIC ESTIMATION ALGORITHM 65next subsections, the reduced measurements are computed using an arbitrary covariance.

We will specify the covariance to use later on.

4.2.3 PHMI FORMULA

Here we develop an expression for the PHMI that will be used to design the algorithm.

The PHMI is defined as:

( )PHMI=Prob true estI I− > ∆

Because we do not know what the parameter u is at a given time, let us assign a density to

it, p(u). The formula for PHMI can then be written:

( ) ( ) ( )0

Prob HMI = Prob HMI|u

u

u p u du=+∞

=∫

Ideally we would like to have the PHMI below the requirement for any distribution of u.

So the new requirement is:

( ) 8Prob HMI| 2.25 10 for any u u−≤

It is therefore a whole curve that must be under the critical value. Let us now develop the

expression for , where we use an inflation factor to multiply the nominal

covariance of the process noise. For this we need to introduce some additional notation.

The estimation variance using a set of coefficients λ and assuming a process noise u can be

written as we did above:

(Prob HMI|u)

( ) ( ) ( )2 2 2 2,0,est process measu uσ λ σ λ σ= + λ

where the first term corresponds to the error due to the process noise and the second one

corresponds to the measurement noise. The expression for each term comes from Equation

(4.8). We have:


( ) ( ) ( ) ( )2,0 0 0 0, 2 , ,T T

process k l kC x x C x x C x xσ λ λ λ λ= − +

( ) ( )2 ,Tmeas k lM x xσ λ λ= λ

Notice that the only assumption on the coefficients here is that the estimate is unbiased.

Let us suppose that we assume an inflation factor, u0, which is a function of the (reduced)

measurements y. Then, given u and y, the PHMI is:

( ) ( )( )( )

0,P HMI| , 2

,est

est

u yu y Q K

uσ λσ λ

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

To obtain the expression for ( )P HMI|u , we need to integrate over all possible y (after the

submission of this work, it was found that this expression is slightly incorrect. However, a

correct treatment of the problem has very little impact on the final inflation factor as

derived by this equation (see Appendix A.4)):

( ) ( ) ( )P HMI| P HMI| , |y

u u y p y= ∫ u dy

We can write the final expression for ( )P HMI|u by replacing the different terms by their

analytical expressions:

( ) ( )( )( ) ( ) ( )

( ) 1

0 21322

, 1P HMI| 2, 2

Ty S u yest

nesty

u yu Q K e

u S u

σ λσ λ π

−

−

−

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠∫ dy (4.19)

Here S(u) is the covariance of the reduced measurements; it is equal to:

( ) ( ) ( ) ( ). .2 2 20 0 0

red redT T TS u u C M u C M u C M= Γ + Γ = Γ Γ +Γ Γ = +

We can at this point formulate the problem of finding the function and the

coefficients λ as follows:

( )0u y


Minimize ( )( )( )20,estE u yσ λ

Subject to ( ) -8P HMI| 2.25 10u ≤ for any u (4.20)

We now have an analytic formulation of the problem. However, it is still a difficult

problem. Not only does the constraint hold for a continuum of parameters, but the

evaluation of (4.20) involves an n-3-fold integral, where the typical size of n is 30. In

addition, the problem of finding the coefficients is not independent of the search of u0.

4.2.4 CASE WITH NO MEASUREMENT NOISE

We now consider the case where there is no measurement noise: M=0. This case is simpler

so it will give us further insight. In addition, the calculations introduced here will be used

in the general case. We have C=u2C0 and 20W u C 1− −= . Let us suppose that we compute

the matrix Г based on C0. Then the reduced residuals y are zero mean and their covariance

is . Expression (4.20) for 23nu I − ( )P HMI|u is simpler now:

( ) ( )

( )

20 233

22

1P HMI| 22

Ty yu

nny

u yu Q K e

u uπ

−

−−

⎛ ⎞= ⎜ ⎟

⎝ ⎠∫ dy (4.21)

This expression does not depend on the coefficients λ: the determination of the optimal

coefficients λ is independent from the search for u0. Now, based on the remarks made in

the first subsection, we are going to limit the search of u0 to a positive definite quadratic

form of y:

( )20

Tu y y Ry=

This is a fundamental assumption for this whole section. We further simplify (4.21) by

performing the change of variablesuz y= :


( ) ( )( )

23

2

1P HMI| 22

Tz zT

nz

u Q K z Rz eπ

−

−= ∫ dz (4.22)

This expression does not depend on u. We can now apply Craig’s formula for the Q-

function [Craig], which is:

( ) ( )

22

222sin2

0

1 12

ts

tQ t e ds e d

π

φ

φ

φππ

−+∞ −

=

= =∫ ∫

The advantage of this formula, which is extensively used in communication theory, is that

the integration limits do not depend on the argument of Q. Replacing the expression in

(4.22) we get:

( ) ( )

( )

222

2sin 23

0 2

1 1P HMI| 22

TTz Rz z zK

nz

u e d e

π

φ

φ

φπ π

− −

−=

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

∫ ∫ dz

If we now reverse the order of integration we get:

( ) ( )

( )

222

2sin 23

0 2

2 1P HMI|2

TTz Rz z zK

nz

u e e d

π

φ

φ

zdφπ π

− −

−=

= ∫ ∫

The argument of the one-dimensional integral is:

( )

( ) ( )( )

2 232 2

12 sin2sin 2

3 32 2

1 1

2 2

TT T

nz Rz Rz zK z K I z

n nz z

e e dz e φφ

π π

−

⎛ ⎞− ⎜− +− ⎜ ⎟

⎝ ⎠− −=∫ ∫ dz

⎟

We can find a closed form for this integral by noticing that:

( ) ( )( )

232

11

22 sin2

3 322

1 1sin 2

Tn

Rz K I z

n nz

RK I e dφ

φ π

−

⎛ ⎞⎜ ⎟− +⎜ ⎟⎝ ⎠

− − z+ =∫

IONOSPHERIC ESTIMATION ALGORITHM 69This is true because we are integrating a Gaussian distribution density over the whole

space. The final result is:

( ) ( )

12 2

232

0

2P HMI|sin n

Ru K I

π

φ

dφπ φ

−

−=

= +∫ (4.23)

This is now a one-dimensional integral, so it is possible to evaluate it. Recall that we want

to:

Minimize ( )TE z Rz subject to (4.23)

Although we will not do it here, it is possible to show that the best R is a multiple of the

identity (the proof uses the fact that to minimize a sum of positive real numbers such that

the product is constant we need to have all numbers equal). As a result, the best inflation

factor in the case of no measurement noise is a multiple of the chi-square statistic:

( )20

Tu y y yα=

The equation to determine α is given by (4.23) and can be written:

( )

32 2

-8 22

0

22.25*10 1sin

n

K

π

φ

α dφπ φ

−−

=

⎛ ⎞= +⎜ ⎟⎜ ⎟

⎝ ⎠∫ (4.24)

The result only depends on the number of measurements n, so it is possible to compute a

table off-line. Let us label each solution αn. Then, in real time, we only need to compute

the chi-square statistic based on the quiet model, C0. Once we have the inflated model, we

can apply the kriging equations. Because the inflated model is a multiple of the nominal

model, it is easy to see from Equation (4.13) that the coefficients are invariant under that

transformation. For the case without measurement noise the problem is solved.

IONOSPHERIC ESTIMATION ALGORITHM 704.2.5 GENERAL CASE (WITH MEASUREMENT NOISE)

The case with measurement noise is more difficult. We need to consider the PHMI

Equation (4.19) and transform the problem so it is treatable. Again, we look for a function

. As we did in the previous case, we are only going to consider quadratic forms. If

we do so, we can apply Craig’s formula for the Q-function again and reverse the order of

integration.

( )20u y

We first want to get rid of the coefficients λ. We do this by noticing that:

( )( )( )

( )0 0,,

est

est

u y u yQ K Q K

u uσ λσ λ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

∼

More precisely, we have:

( )( )( )

( )0 0,,

est

est

u y u yQ K Q K

u uσ λσ λ

⎛ ⎞ ⎛ ⎞≤⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

for ( )0u y u≤

and for we have: ( )0u y u≥

( )( )( )

( )0 0,,

est

est

u y u yQ K Q K

u uσ λσ λ

⎛ ⎞ ⎛ ⎞≥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

which is a problem. However, in that case we also have:

( )( )( )

0 -8,2.5 10

,est

est

u yQ K

uσ λσ λ

⎛ ⎞≤⎜ ⎟⎜ ⎟

⎝ ⎠

This indicates that for the contribution of the integral (4.19) will not be a large

one. We could take into account this approximation by simply inflating K from 5.592 to

5.69 (2%). The proof is straightforward but tedious. However, this approximation has

( )0u y u≥


consistently proven to be a conservative one, due to the fact that ( ) ( )2 2,0meas processσ λ σ λ -

as we saw in Subsection 4.1.5. After this approximation, we have:

( ) ( )

( ) ( )

( ) 1

0 21322

1P HMI| 22

Ty S u y

ny

u yu Q K e

u S uπ

−

−

−

⎛ ⎞= ⎜ ⎟

⎝ ⎠∫ dy (4.25)

Expression (4.25) is a big step because we have decorrelated the search for the inflation

factor from the search for the coefficients.

By writing that and applying Craig’s formula in the same way we did in the

previous subsection, we obtain:

( )20

Tu y y Ry=

( ) ( )( )

12

22

32 20

2P HMI|sin n

RS uu K I

u

π

φ

dφπ φ

−

−=

= +∫ (4.26)

where, as we saw earlier, ( ) ( ) ( ).2 20 0

red redT TS u u C M u C M= Γ Γ +Γ Γ = + . . As we did for the

noiseless case, we have reduced an n-3-fold integral to a one-dimensional integral. Now

we can evaluate this function of u for a given R, and make sure that it is always below the

critical PHMI value. To gain some insight on the behavior of ( )P HMI|u as expressed in

(4.26), we now study an example.

4.2.6 EXAMPLE

Let us suppose that ( ).0 3

rednC Iβ −= and ( ).

3red

nM Iη −= . The factor β reflects the relative

magnitude of the process noise versus the measurement noise in the nominal covariance

model. This example can be treated easily (there is no need to compute determinants) and

it is important since we will be able to reduce the general case to it. Because of the

definition of these matrices, we have β+η=1. Because it is optimal in the case without

measurement noise, it is reasonable to take 3nR Iα −= . This means that the inflation factor

IONOSPHERIC ESTIMATION ALGORITHM 72is a multiple of the chi-square statistic computed assuming the nominal covariance model.

Equation (4.26) is now:

( )( )( )( )

1222

3 3232 2

0

12P HMI|sin

n nn

u I Iu K I

u

π

φ

α β βdφ

π φ

−

− −−

=

+ −= +∫

Since the matrix in the integral is diagonal, this equation is simplified:

( )( )( )( )

32 22

22 2

0

12P HMI| 1sin

n

uu K

u

π

φ

α β βdφ

π φ

−−

=

⎛ ⎞+ −⎜=⎜⎝ ⎠∫ ⎟+

⎟ (4.27)

The difference now is that there is no determinant in the term to be integrated, only a

product. We can now study the behavior of ( )P HMI|u as a function of u, parameterized

on the relative contribution of the process noise, β. Figure 4.3 shows several of these

curves for different values of β in the case n=20. The parameter α was computed using

Equation (4.24).


0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

-9

10-8

10-7

10-6

10-5

10-4

PH

MI

u

β = .25β = .50β = 1

Figure 4.3: PHMI as a function of u parameterized by β

The blue curve corresponds to the case of pure process noise. As expected, it does not

depend on the parameter u. As we said, P(HMI|u) increases with u. It increases faster as

the measurement noise becomes larger compared to the nominal process noise.

This function has two important properties. It increases with u and has a limit as u goes to

infinity. It is straightforward to compute this limit:

( ) ( )

32 2

22

0

2P HMI|+ 1sin

n

K

π

φ

αβ dφπ φ

−−

=

⎛ ⎞∞ = +⎜ ⎟⎜ ⎟

⎝ ⎠∫ (4.28)

This formula is identical to (4.24) except for the factor β. It suggests that to get a safe

inflation factor, we could compute the table as we did in the case with no measurement

noise, and then scale the result by the inverse of β.

IONOSPHERIC ESTIMATION ALGORITHM 74The behavior of P(HMI|u) is due to the fact that when we compute the chi-square statistic,

we assume a certain ratio between process noise and measurement noise. However, if the

ratio is in reality larger than assumed, the inflation factor is going to act as if some of the

variability that is really due to the process noise is due to the measurement noise, thus

underestimating the process noise. That would not be a problem if the ratio of the

contribution of process noise to the measurement noise in the estimation variance was the

same as in the ratio between the covariances. But this is not the case. As we saw in

Section 4.1.5, the term coming from the measurement noise is much smaller, because the

large number of measurements allows us to average it down. What this section shows is

that the ratio between process noise and measurement noise is critical. We have also

shown that when the reduced matrices are proportional to the identity, it is possible to

compute an inflation factor safe for all u.

4.2.7 RATIO BETWEEN PROCESS NOISE AND MEASUREMENT NOISE

In this section we are going to show that by using the general case, we can find an upper

bound of the PHMI where the matrices are proportional to the identity (like in the previous

section). Also, we are going to evaluate the ratio between process noise and measurement

noise in the real system.

Because it is optimal in the noiseless case, we take again 3nR Iα −= . The PHMI condition

is:

( )( ) ( )

( )

12

. .222 0

32 20

2P HMI|sin

red red

nu C Mu K I

u

π

φ

dα φπ φ

−

−=

+= +∫

We remind the reader that we have ( ) ( ). .0

red rednC M I 3−+ = . There is no reason for each term

in the sum to be a multiple of the identity. However, we can suppose without loss of

generality that both matrices are diagonal (to see this, diagonalize one of them; the other

one needs to be diagonal in the same basis). Let us write:

IONOSPHERIC ESTIMATION ALGORITHM 75( ) [ ]( ).0 1

rednC diag c c −= 3 and ( ) [ ]( ).

1 3red

nM diag m m −=

By construction, and . We have: 1i ic m+ = , 0i ic m ≥

( ) ( )

122 3 2

22 2

10

2P HMI| 1sin

ni i

i

u c mu Ku

π

φ

dα φπ φ

−−

==

⎛ ⎞+= +⎜ ⎟⎜ ⎟

⎝ ⎠∏∫

It is possible to show that:

( ) ( )

31 1 2

3 33 32

21 12

2 20

2P HMI| 1sin

n

n nn n

i ii i

u c mu K

u

π

φ

dα φπ φ

−−

− −− −

= =

=

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟+⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠≤ +⎜ ⎟

⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∏ ∏∫ (4.29)

This inequality is based on the following general inequality (see Appendix A for proof):

( )1

1 1

1 1 kb

pp pp

kk k

b= =

⎛ ⎞⎛ ⎞⎜ ⎟+ ≤ +⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎝ ⎠∏ ∏ 0k ≥ for b (4.30)

The upper bound in (4.29) of the PHMI is similar to (4.27) with ( )

113 3

30

1

n n red ni

i

c Cβ− −

−

=

⎛ ⎞= =⎜ ⎟⎝ ⎠∏ .

When the matrices are a multiple of the identity matrix, there is equality. The closer these

matrices are to a multiple of the identity, the tighter the bound is. Let us now examine the

empirical distribution of the parameter β in the system. It will be a function of the

covariance of the measurement noise and the geometry of the measurements (we assume

the nominal model for the process noise covariance). Figure 4.4 shows the histogram of

the computed values of β for July 2, 2000.


0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650

50

100

150

200

250

300

350

400

450

500

β

Figure 4.4: Process noise contribution

At this point, we could choose to pick a fixed value for β or to compute it in real time. A

fixed value would need to be a lower bound of all possible values of β. In the final

algorithm we will compute β in real time. This way we only multiply by what is needed in

a particular situation. The computation of the parameter β is not as expensive as this proof

might suggest. We have:

( ) ( ) ( )( )( )30

0 0i i

redn T T T Ti iC C C C W WG GWG GWβ −

≠ ≠

= = Γ Γ = Γ Γ = −∏ ∏

In this expression, designates the eigenvalues of a matrix. The computation of the last

expression only requires finding the eigenvalues of ( )( )T TC W WG GWG GW− . In this

expression, C is taken to be the nominal covariance, C0, and W is computed based on C0.

IONOSPHERIC ESTIMATION ALGORITHM 774.2.8 ADDING A THRESHOLD

In the previous sections we have taken the inflation factor to be a quadratic form of the

measurement, and showed that we could compute an inflation factor that satisfies the

PHMI. The quadratic form was chosen for two reasons. It is a good choice in the

measurement noise free case and it allows us to obtain a relatively simple analytical form

of the PHMI. However, the curves in Figure 4.3 suggest that we would like the inflation

factor to increase faster than a quadratic as the residuals increase. It is an open problem to

find a functional form for ( )20u y that is close to quadratic but increases faster, and that

makes the integral (4.22) easily computable. Another solution consists of adding a

threshold to the chi-square statistic. If the chi-square statistic is larger than the threshold,

the error bound is set to a maximum value (the maximum ever observed) [ADD],

[Walter00]. The idea behind this is as follows. The PHMI becomes larger than desired for

large values of u. These values of u are likely to result in a large chi-square statistic. If we

only accept chi-square statistics below a certain threshold, we will be less likely to incur in

HMI. In this section we study the effect of adding a threshold on the PHMI. The analysis

done in this section relies on two unproven approximations. Despite the lack of proof,

these two approximations are intuitively correct.

The new PHMI formula is similar to (4.26). Let us call T the threshold:

( ) ( )

( ) ( )

( ) 1

0 21322

1P HMI| 22

T

T

y S u y

ny y T

u yu Q K e

u S uπ

−

−

−≤

⎛ ⎞= ⎜ ⎟

⎝ ⎠∫ dy

The only difference lies in the integration domain. Now we only consider y inside the

ellipsoid . This change again makes the computation of the PHMI difficult and

unpractical (because there is no formula to integrate the density of a multivariate Gaussian

distribution over an arbitrary ellipsoid). Instead, we are going to rely on the following

approximation:

Ty y T=


( ) ( ) ( ) ( ) ( )0 02 | 2 |T Tyy y T y y T

u y u yQ K p y u dy Q K p y u dy p y u dy

u u≤ ≤

|⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞

= ⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

∫ ∫ ∫

This approximation is obtained by assuming that:

( ) ( ) ( ) ( )( )0 01 1T Tu y u yE Q K y y T E Q K E y y T

u u⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞

≤ = ≤⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

where is a function equal to one over the ellipsoid and zero otherwise. Now

we can use the results obtained in the previous section. We know how to find a

computable upper bound of the first term (Subsection 4.2.7), so we turn our attention to the

other term:

(1 Ty y T≤ )

( )( ) ( )

( ) 1

21322

1Prob2

T

T

y S u yT

ny y T

y y T e dyS uπ

−

−

−≤

≤ = ∫

There is no known analytical solution for this integral. Even worse, the numerical

integration for such dimensions is not trivial. We will first show the link from this integral

to a weighted sum of chi-squares, which is a problem which has been studied [Gabler]. Let

us do the change of variables ( )12z S u y−= . We have:

( )( )( )

23

2

1Prob2

T

T

z zT

nz S u z T

y y T e dzπ

−

−≤

≤ = ∫ (4.31)

As we said earlier, S(u) can be considered to be diagonal without loss of generality:

( ) [ ]( ) [ ]( )21 3 1n nS u u diag c c diag m m− −= + 3

)i+

, so we have:

( ) (3

2 2

1

nT

i ii

z S u z z u c m−

=

=∑ (4.32)

IONOSPHERIC ESTIMATION ALGORITHM 79This shows that (4.31) is the cumulative distribution function of a sum of weighted chi-

squares. In [Gabler], the following approximation is given:

( )1

33 3 32 2 2 2

1 1 1

Prob Probnn n n

i i i i i ii i i

z u c m T z T u c m−−− − −

= = =

⎛ ⎞⎛ ⎞⎛ ⎞ ⎜ ⎟+ ≤ ≈ ≤ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟

⎝ ⎠∑ ∑ ∏

The right term is a chi-square cumulative distribution, so it is easily computed. Now take

one step further by applying the inequality that was used to prove (4.31). We get:

133 33

2 2 21 1

1 1 3 31 3 32

1 1

Prob Probnn nn

i i i ii i n ni n n

i ii i

Tz T u c m z

u c m

−−− −−

= = − −= − −

= =

⎛ ⎞⎜ ⎟⎛ ⎞ ⎜ ⎟⎛ ⎞⎜ ⎟≤ + ≤ ≤⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛ ⎞⎝ ⎠ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∑ ∑∏∏ ∏

In order to get an idea of the benefit the chi-square yields, we apply these results to the

example studied in subsection 4.2.6: ic β= , 1im β= − . The threshold was chosen such

that:

32 3

1

Prob 10n

ii

z T−

−

=

⎛ ⎞≤ ≤⎜ ⎟

⎝ ⎠∑

This is the probability of false alarm if we are in the nominal state, but here it is not a

specification of the algorithm. Figure 4.5 shows the new PHMI curves generated by

adding the threshold to the algorithm. We can see that now the PHMI is no longer an

increasing function of u, and that it has a maximum value. This maximum value needs to

be below the design requirement (2.25X10-8). We see that despite the threshold, the PHMI

is unacceptably large and that there is little benefit compared to the situation without the

threshold.


0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

PH

MI

u

β = .25β = .50β = 1

Figure 4.5: PHMI as a function of u when a threshold is added

There are two ways of bringing down the curves: by decreasing the threshold T or by

increasing the inflation factor α. Decreasing the threshold is not an attractive option,

because when the ionosphere is well behaved (i.e. when the nominal covariance is an upper

bound of the true covariance) we want the probability of false alarm to be very small. As a

result, we will have to rely only on the inflation factor. None of the analysis done in this

subsection will be used to generate the inflation factor because it relies on some unproven

approximations and the benefit seems to be small. The conclusion of this subsection is

that, within our model, adding a threshold does not help much. However, the

implementation of the threshold will be important when we deal with the irregularities that

are not well described by the assumed model, as we will see later.

IONOSPHERIC ESTIMATION ALGORITHM 814.2.9 DETERMINATION OF THE INFLATION FACTOR

The purpose of this section was to determine an algorithm that computes an inflation factor

u0 such that the model u02C0 can be used as if there was no uncertainty. This subsection

summarizes the different steps in the computation, and distinguishes what can be done

offline from what must be done in real time.

The inflation factor is given by ( )20

Tu y y yα= , where we have:

( )( )1

0 0 0 0T T T T

meas measy y I W W G G W G G W I−

= −

where . We still need to determine α. ( 10 0W C M −= + )

Subsection 4.2.7 showed that:

( ) ( )

32 2

22

0

2P HMI| 1sin

n

u K

π

φ

αβ dφπ φ

−−

=

⎛ ⎞≤ +⎜ ⎟⎜ ⎟

⎝ ⎠∫ where ( )

113 3

30

1

n n red ni

i

c Cβ− −

−

=

⎛ ⎞= =⎜ ⎟⎝ ⎠∏ (4.33)

The parameter β is given by:

( )( )( )1

31

0 0 0 0 00i

nT T

i C W W G GW G GWλ

β−−

≠

⎡ ⎤= −⎢ ⎥⎣ ⎦∏ (4.34)

In Subsection 4.2.4, we determined for a given n a value αn such that:

( )

32 2

-8 22

0

22.25 *10 1sin

n

nK d

π

φ

α φπ φ

−−

=

⎛ ⎞= +⎜ ⎟⎜ ⎟

⎝ ⎠∫

According to Equation (4.33), if we choose α such that: nαβ α= , we will have:


( ) ( ) ( )

3 32 22 2

2 22 2

0 0

2 2P HMI| 1 1 2.25 *10sin sin

n n

nu K d K

π π

φ φ

ααβ φπ φ π φ

− −− −

= =

⎛ ⎞ ⎛ ⎞≤ + = + =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫ ∫ -8

Once we have the inflation factor, we proceed as if the covariance model was the inflated

model and not worry anymore about the model uncertainty. To summarize, for a given set

of measurements we have:

( )( )120 0 0 0 0

T T Tnmeas measu I W W G G W G G W Iα

β−

= −

where β is given by Equation (4.33) and αn is read in a table. Figure 4.6 shows the flow

chart of the real time estimation of the inflation factor. The different variables were

defined at the beginning of the chapter.

Imeas χ2 =ImeasTP0Imeas

Figure 4.6: Real time inflation factor determination

Once we have this inflation factor, we use the process noise whose covariance is C=u02C0.

Using this covariance, we can now apply the kriging method outlined in the first section to

compute the coefficients and the estimation variance.

4.3 THE THREAT MODEL

Section (4.1) described how to compute the kriging estimate assuming a certain model

(known u). Section (4.2) showed how to take into account the model uncertainty by

inflating the nominal model by u0. The two previous sections have thus determined the

Compute C0

W0=(C0+M)-1

P0=W0-W0G(GTWG)-1GTW0 β=function(C0,P0)

(Equation (4.33))

Look up αn in table

u02= αn β-1 χ2 G, M

n

IONOSPHERIC ESTIMATION ALGORITHM 83ionospheric estimation algorithm as if the model (parameterized by u) derived in Chapter 3

always held. However, as we saw at the end of Chapter 3, there is evidence that the model

might not always hold: large gradients sometimes divide quiet regions (low u) from

disturbed regions (large u). The biggest threat to the system can happen when the

measurements are all made in the quiet region, so the system is blind to the coming

gradient. As this is an extremely difficult situation to model in probabilistic terms, the

method followed here (and used in the current ionospheric estimation algorithm for

WAAS) relies on the post-processed data and data deprivation schemes of real data. An

extensive description of this algorithm is presented in [Sparks01].

The net effect of this part of the algorithm, called threat model, is twofold. The first part to

adds an additional term to the error bound that is a function of the geometry of the

measurements surrounding the region that is estimated. The second part conditions the

computation of an error bound to have the chi-square statistic be below a threshold, as it

was outlined earlier. The reason for implementing the chi-square threshold is the fact that

violations of stationarity occur much more often during disturbed periods. The idea is to

test a given algorithm separating the real data in data used for estimation from the test data

(which simulates a user). Then, it is checked whether the error bound computed with the

estimation algorithm gives an upper bound of the estimation error. If the error is larger

than the estimated error bound, we assume that for a situation with a similar geometry as

the one tested, we will need to increase its error bound by as much as the real error outgrew

the estimated error bound. In the current WAAS algorithm, the geometry is characterized

by two parameters: the number of measurements used for the correction and the distance

from the center of gravity of the measurements to the location to be estimated [Sparks01].

In this thesis, we will not give a full account of the threat model because it is still an

evolving part of the algorithm and because it can be considered an external part of the

algorithm. Furthermore, new ways of computing the threat model show that it is possible

to minimize the effect of the threat model [Sparks03]. In Chapter 6, performance will be

presented with and without the threat model.

Chapter 5

Adaptation of the Algorithm to the WAAS Message

Now we have a way of computing at any CONUS location of the thin shell an estimate of

the vertical ionospheric delay and an error bound. However, this algorithm assumes that

the user has all the measurement information. Because of bandwidth limitations and the

need to reduce as much as possible the computational load on the user, the ionospheric

correction message is in a format described in the WAAS Minimum Operational Standard

(MOPS) [MOPS]. This document specifies that the ionospheric information be sent in a

grid of 5 by 5 degrees or 10 by 10 degrees in the thin shell at a height of 350 km. At each

ionospheric grid point (IGP), the user receives a vertical ionospheric grid delay (IGD) and a

grid ionospheric vertical error (GIVE). The user calculates each of the ionospheric delays

corresponding to the satellites in sight as well as the confidence bounds from this grid,

according to an algorithm which is also set in the WAAS MOPS. Figure 5.1 shows the set

of grid points as circles. To be applicable in the short term, an estimation algorithm must

be adapted to the ionospheric grid.

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 85

conf

iden

ce b

ound

(m) -

95%

0.49 0.97 1.46 1.94 2.43 2.92 3.4 3.89 4.37 4.86 5.83 7.29 9.72 24.3 72.9 NM

-180 -160 -140 -120 -100 -80 -6010

20

30

40

50

60

70

Longitude (deg)

Latit

ude

(deg

)

Figure 5.1: Ionospheric grid points as specified in the MOPS

The algorithm as it has been outlined previously cannot be applied directly. Figure 5.2

illustrates the problem in a two-dimensional example. In this diagram, the blue dots

represent the IPP measurement location on the horizontal and the value of the vertical delay

on the vertical. The estimate computed using kriging at all locations would give a curve

that loosely follows the measurement (the exact shape is determined by the variogram).

With the estimate, there is also a confidence bound, shown in red. As we said, the user

only receives the vertical ionospheric delay at the IGPs, i.e., at the two extremes of the

diagram. The estimate computed by the user (black line) does not coincide with the kriging

estimate. As a result, the error bound sent to the user at the grid points needs to be such

that the interpolation error is accounted for. Fortunately, the situation is not as bad as it is

shown in this diagram, because the grid spacing is not as large compared to the ionospheric

decorrelation, but it still needs to be evaluated. The first decision to make is what vertical

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 86delay to send at the IGP. The most natural choice is to send the value obtained using

kriging at that location, and that is what we will do.

Equation Chapter (Next) Section 5

User confidence bound Confidence bound

Kriging estimateUser estimate

Grid point Grid point

Figure 5.2: Difference between kriging estimate and user estimate

To treat this problem, we will first review how the user forms both the estimate and the

error bound (also called confidence bound). Then we will compute an overbound of the

true user error bound. Finally we will see how to compute the GIVE such that the user

always computes an overbound of the true error bound.

5.1 THE IONOSPHERIC GRID: USER ALGORITHM

For each of its IPPs, the user determines the box of IGPs in which the IPP is contained.

Then the user interpolates the data of the four IGPs which form the box:

4

,1

user i IGP ii

ˆI Iµ=

=∑ (5.1)

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 87Similarly, the user confidence bound, UIVEKσ , is found by interpolating the four

surrounding GIVEσ :

42 2

1UIVE i GIVE

i

σ µσ=

=∑

The µ coefficients are computed by the receiver according to a bilinear interpolation

scheme specified in [MOPS]. This scheme has an essential property that will be used later

in the section. It is unbiased in the sense that:

IGPG Xµ =

The matrix is the geometric matrix corresponding to the IGPs (it is defined as G is for

the measurements). X was defined earlier.

IGPG

5.2 USER ERROR BOUND

The user error bound depends on how the estimate is formed at the IGP by the master

station. Since the kriging estimate is optimal, given the assumed covariance, the WAAS

message will broadcast the delays at the IGP computed using kriging (as described in

Section 5.1.) In this subsection, we will only assume that each IGD estimate is unbiased

(in the sense that the expectation of the estimate over all possible residual fields is the same

as the expectation of the true vertical ionospheric delay). We have:

,ˆ T

IGP i i measI Iλ=

Here λi is the vector of coefficients and measI is the vector of measurements. Since the

estimate formed from the IGPs to the user is also unbiased, we have:

( ) ( )4

,1

ˆ ˆuser i IGP i true

i

E I E I E Iµ=

⎛ ⎞= =⎜ ⎟⎝ ⎠∑

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 88We can therefore calculate the estimation variance. (The reader is reminded that, because

of the inflation factor, we can treat the problem as if there was no model uncertainty.) We

have:

( )( ) ( ) ( ) ( ) ( )( )4 42

1 1

ˆ , 2 , , ,T

Tuser true i i k i i l k l k i i

i iE I I C x x C x x C x x M x x

4

1iµ λ µ λ

= =

⎛ ⎞ ⎛ ⎞− = − + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∑ ∑ µ λ

=∑

This is the expression for the true estimation variance at the user IPP when interpolating the

four surrounding IGPs (although it would be true for any number of IGPs).

5.3 GIVE COMPUTATION

In this section we present the method to compute the GIVE at each IGP. We will assume

that at each grid point, the vertical ionospheric delay broadcast is the kriging estimate

obtained by the algorithm outlined in Chapter 4. After showing the principle of the

calculation, we will give the details.

5.3.1 PRINCIPLE

The idea of the GIVE computation is to consider each quadrant, that is, each group of four

IGPs, and to find four quantities, σi (one for each IGP), such that for any user IPP located in

the quadrant:

( )42 2

1user true i i

i

E I I µσ=

− ≤∑ (5.2)


σ1(A) σ2

(A) σ2(B) σ3

(B)

A B

σ5(A) σ6

(B)σ4(A) σ5

(B)

σ4(C) σ5

(C) σ5(D) σ6

(D)

C D

σ7(C) σ8

(C) σ9(D)σ8

(D)

Figure 5.3: GIVE computation

For each quadrant, we will assign a σi to each of the four IGPs. Since each IGP belongs to

four quadrants, we will have 4 different σi for each IGP. σGIVE2 will be the maximum of

these four quantities. Figure 5.3 illustrates the situation. This diagram shows nine IGPs,

and for each quadrant, the four σi corresponding to Equation (5.2). Here, we want to

compute σGIVE2 at the IGP in the center (Number 5). First, for each quadrant, A, B, C and

D, we compute the four σi. Because we want the broadcast value at IGP number 5 to be

safe for any user having an IPP in A, B, C or D, we will need to have:

( ) ( ) ( ) ( )( )5 5 5 5max , , ,A B C DGIVEσ σ σ σ= σ

,

5.3.2 DECOMPOSING THE ESTIMATION VARIANCE IN THREE TERMS

We now focus on getting the four σi for each quadrant. In order to find these four terms,

we write:

( ) ( ) ( ) ( )4 42

1 1

ˆ , 2 ,T Tuser true i i i i k k

i i

E I I C x x S C x x C x xµ µ µ λ µ= =

⎛ ⎞− = + + −⎜ ⎟⎝ ⎠

∑ ∑ (5.3)

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 90The expression for S is:

( ) ( )( ) ( ) ( ), , , , TT Tk l k l i l i lS C x x M x x C x x C x x= Λ + Λ −Λ − Λ

The matrix is formed of the coefficients used to estimate each delay at the IGP: Λ

[ ]1 4λ λΛ =

If the expression given by (5.3) was affine in µ, the problem would be solved. But it is not:

the second term is quadratic, and the third has an even more complicated dependency on µ.

The problem now is to find an upper bound linear in µ for each one of the three terms. For

the first term, we have:

( ) (4

1, ,i i i

i

C x x C x xµ=

=∑ )

This is the case because we know that the components of the vector µ have a sum of one

and C(x,x) does not depend on the location. The other two terms require more work.

5.3.3 BOUNDING THE QUADRATIC TERM

The second term is quadratic in µ. Let us assume for now that we can write ( )( )S S S −+= + ,

where the first term is a positive semi-definite matrix and the second one is a negative

semi-definite matrix. Because of the convexity of positive semi-definite quadratic forms it

is easy to see that:

( ) ( )4

1

Ti ii

i

S Sµ µ µ+ +

=

≤∑

For the definite negative term, we use the fact that a tangent plane of the graph is an upper

bound. For any µ0 we have:

( ) ( ) ( )0 0 0T Sµ µ µ µ−− − ≤

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 91After developing the equation we have:

( ) ( ) ( )0 0 02T T TS S S Tbµ µ µ µ µ µ µ− − −≤ − =

The choice of the point µ0 is arbitrary for each quadrant. We will take µ0=[1 1 1 1]/4.

Figure 5.4 illustrates what is done in the case of a definite negative quadratic form. In blue

we have the negative quadratic form, in red the tangent at µ0.

IGP,2IGP,1 µ0

Figure 5.4: Bounding the definite negative term

How do we find the decomposition ( )( )S S S −+= + ? We first show a general method and

then show that when we use the kriging estimate at each grid point, there is a natural

decomposition. For the general method, we find D diagonal and V orthogonal such

that . We then write that TS V DV= ( ) ( )D D D+ −= + where the first term has all the

positive eigenvalues and the second one the negative ones. We then have ( )( ) TS V

and

D V++ =( )( ) TS V D −− = V . Notice that S is only a 4 by 4 symmetric matrix, so this computation is

easy to realize in real time. This computation is valid for coefficients λ, such that the

estimate is unbiased. There is no need to assume that the coefficients are computed using

kriging.

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 92For the case in which the coefficients are computed using the kriging method using the

same inflated covariance, there is a natural decomposition that does not require the search

for eigenvalues. We have:

( ) ( ) ( ) ( ) ( )1, , ,

T TT T Ti i k i i k i k i kS X G WC x x G WG X G WC x x C x x WC x x

−⎡ ⎤ ⎡ ⎤= − − −⎣ ⎦ ⎣ ⎦ ,

It is clear that the first term is positive semi-definite and the second one is negative semi-

definite.

5.3.4 BOUNDING THE LAST TERM

The last term is the most problematic due to its dependence on the covariance from the IPP

user location to the IPP measurements, which is not known by the user. The idea here is to

find an upper bound for:

( ) ( )4

1

, ,Ti j j k k

j

C x x C x xλ µ=

⎛ ⎞−⎜ ⎟

⎝ ⎠∑

We first show that there exists ε and δ positive such that:

( ) ( )4

1, ,j j k k

jC x x C x xδ µ

=

− ≤ − ≤∑ ε (5.4)

These inequalities exploit the particular shape of the covariance (the expression for it was

given in Chapter 3.) We have:

( ) ( ) 2 max0, ,j k k

dC x x C x x u ca

δ− ≥ − = −

where is the maximum distance between maxd ix and kx , which is smaller than 700 km in

the case of a 5 by 5 grid. This inequality is simply the fact that the difference between two

values of a function is smaller than the maximum of the derivative times the distance

between the two values.

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 93For the other inequality, we use the fact that within a quadrant, the covariance from a

measurement to any other point in the quadrant is very close to concave. This function has

the shape of an inverted cone whose vertex is at kx and whose generator is:

( ) 20

daC d u ce

−=

for d positive. This function is almost affine over a distance of 700 km (since a=32000

km). As a result, it is easy to find – but cumbersome to write – a concave function such that

within the four IGPs the following is true:

( ) ( )0 ' , ,k kC x x C x x ε≤ − ≤

with ε very small (a value is given in Appendix A). Figure 5.5 illustrates how to define C’

and the resulting ε.

C’(xk,x)

ε

C(xk,x)

xk

Figure 5.5: Diagram of the covariance and a concave approximation

Since 4

1j j

jx xµ

=

=∑ and C’ is concave we have:

( ) ( )4

1

' , ' ,j j k kj

C x x C x xµ=

0− ≤∑

As a result we have:


( ) ( )4

1, ,j j k k

iC x x C x xµ ε

=

− ≤∑

Now we can write that:

( ) ( )4

1 0, ,

ik ik

Ti j j k k ik ik

jC x x C x x

λ λ 0iλ µ δ ε

= <

⎛ ⎞η

>

− ≤ Λ + Λ =⎜ ⎟⎝ ⎠∑ ∑ ∑

4

1i

We finally have:

( ) ( ) ( )4 4 42

1 1 1

ˆ ,user true i i i i ii i i i ii i i

E I I C x x S bµ µ µ+

= = =

− ≤ + + + µη=

∑ ∑ ∑ ∑

This means that we can take:

( ) ( )2 ,i i i ii iC x x S b iσ η+= + + + (5.5)

As said earlier, for each IGP there are four possible 2iσ corresponding to each quadrant

where the IGP has influence (A, B, C and D). The computed will be the maximum

of these four values.

2GIVEσ

5.3.5 GIVE COMPUTATION: SUMMARY

We summarize here the steps that are necessary to compute the GIVE for a given IGP (let

us index it by i). First, we determine the quadrants that are influenced by IGPi. Usually,

this includes the four 5 by 5 degree quadrants that have IGPi as a common edge (However,

at the edge of coverage the pattern will be slightly more complicated). For each of these

regions, we need to compute all 2iσ associated with IGPi, ( )2

i Aσ , ( )2i Bσ , ( )2

i Cσ , and

. There are therefore four calculations to complete, one for each quadrant. ( )2i Dσ

The first and fourth terms in (5.5) are common to all four of them, so they do not need to be

computed 4 times. For the second and third term we need to compute the matrix S for each

quadrant. For this purpose we need to determine the matrix Λ for each quadrant, that is, the

ADAPTATION OF THE WAAS MESSAGE TO THE ALGORITHM 95kriging coefficients at each one of the eight surrounding IGPs. Once we have W, this step

only involves matrix products.

Imeas, G, M, Xi, C0, XIGP (for all surrounding IGPs)

C=u02C0 (see Figure 4.6)

Figure 5.6: GIVE computation flowchart

Figure 5.6 summarizes the calculation of the GIVE. This flow chart is valid for one IGP:

first the inflation factor is computed (see Chapter 4); Once this done, we compute the

kriging coefficients corresponding to the IGP considered, the term Sii(+), C(x,x), and the

term bi. We sum them and add the threat model term, which only depends on the geometry

of the IPPs.

Compute λi at IGP using covariance C

C(x,x)=u02C0(0,0)

( ) ( )

0 0k k

k ki i i

λ λ

η δ λ ε λ≤ ≥

= +∑ ∑

Compute S(-) for

each quadrant

where the IGP belongs

and compute bi, q for

( )iiS +

each quadrant q.

bi=max(bi,q)

Compute threat model term:

σthreat

2= function of G

( ) ( )2 ,i i i ii i iC x x S bσ η+= + + +

2 2 2,GIVE i i threatσ σ σ= +

Chapter 6

Performance Analysis

For a safety critical system like WAAS or any other satellite based augmentation system

whose main purpose is to provide integrity to users, the first requirement is the ability to

give a hard bound on the estimation error. For this reason, the first thing to check in the

ionospheric estimation algorithm is that the computed error bounds are really bounding the

possible estimation error. Only when we are confident that the error bounds are always an

upper bound of the inaccuracy can we begin to evaluate the benefits of the algorithm in

terms of availability.

This chapter starts by checking the integrity of the algorithm outlined in Chapter 5. For

this purpose we will describe the different possible schemes to test integrity, which are all

based in cross-validation and data deprivation [Sparks01]. Once we have shown that the

estimation algorithm is safe (within reasonable data deprivation schemes), we will explain

the different metrics that are commonly used to measure the performance of an SBAS.

We will continue by introducing the Matlab Algorithm Availability Simulation Tool

(MAAST), the service volume analysis tool used to evaluate the performance of the

ionospheric estimation algorithm once it is integrated in the system. Then, the results of

the MAAST simulations will be plotted and compared to the simulations generated using

the current WAAS ionospheric estimation algorithm. In the last section we evaluate the

performance of the algorithm without considering the bandwidth limitations.

PERFORMANCE ANALYSIS 976.1 INTEGRITY CHECK

In order to check the integrity of the algorithm offline, we want to test whether the

computed error bounds are larger than the actual error. As explained in Chapter 2, the

requirement is more precise than that, and we will go back to it. There is only one way to

do this, which is by using cross-validation [Hastie]. Cross-validation is widely used for

estimating prediction error. Here we use it to test whether the predicted error is larger than

the actual error. The idea is to put aside a validation set and use it to assess the capacity of

the algorithm to produce safe error bounds. That is, we compute the estimate and the error

bounds based on the set of remaining measurements. To be exhaustive, the validation set

should be any possible set (any subset of the measurements should be tested as a validation

set). The large amount of data make this approach infeasible, so we limit the cross-

validation to certain types of sets. First we will consider leave-one-out cross-validation or

single cross-validation, which consists of excluding each measurement one at a time and

performing the estimation with the remaining measurements. Then we will consider a

more severe data-deprivation scheme.

The data used for these integrity checks is the post-processed data corresponding to sixteen

quiet and stormy days over CONUS (February 12, 2000; March 26, 2000; April 6-8, 2000;

May 25, 2000; June 6, 2000; July 2, 2000; July 15-16, 2000; March 31, 2001; April 1,

2001; September 4, 7, 8, 11, 2002). Among this set there are many more days

corresponding to stormy days. In reality, the percentage of storm days is small (5%, even

during solar maximum). The total number of measurements tested was 1,592,603. This is

not enough to test our algorithm to 10-7, but there were not more measurements available.

Because we use post-processed data, the measurement noise is assumed to be zero. This

integrity check does not test the final algorithm; rather, it tests that the assumed model

leads to conservative error bounds. In addition, we are not including the effects of the

interpolation error when using the grid (we use the measurements directly to get the kriging

estimate at the location to be estimated) and the threat model term is not included in the

confidence bound (we want to test how well the algorithm would protect the user without

any threat model).

PERFORMANCE ANALYSIS 986.1.1 LEAVE-ONE-OUT CROSS-VALIDATION

In this first section we present the results of single (or leave-one-out) cross-validation. For

each IPP measurement we first select the measurements that are within a certain radius of

the IPP location. (This radius search replicates the search in the real algorithm, and was

explained in subsection 5.1.5). Then, we transform the geodetic coordinates to East-North-

Up coordinates based on the location to be estimated. After that we estimate the inflation

factor by computing the chi-square estimate based on the nominal model and multiplying it

by αn (the table for it was determined in Subsection 5.2.4). Because there is no

measurement noise, we do not need to determine the ratio β (it is one). Once we have the

inflation factor, we can compute the kriging estimate and the error bound from the inflated

model. For the noiseless case, the kriging estimate is the same for any inflation factor

because the matrix C is a multiple of the nominal covariance, C0. This is not the case in

general. In this analysis, the chi-square threshold was not implemented. Figure 6.1 shows

a histogram of the ratio:

est true

est

I Iσ−

In this equation, estI is defined in Equations (4.3) and (4.13), and estσ in Equation (4.14).

We will call this ratio the normalized residual (here the word residual no longer designates

the process noise and instead refers to the difference between true value and estimated

value). The first requirement of the normalized residuals is that they need to always be

below the factor K =5.33. Another requirement, mentioned in Chapter 2, is that the

distribution of the normalized residuals must always be overbound by a unit Gaussian in

the sense that for any s we need to have:

( )2

2ˆ 1Prob 1 2

2

s west true

est s

I I s e dwσ π

−

−

⎛ ⎞−≤ ≤ = −⎜ ⎟⎜ ⎟

⎝ ⎠∫ Q s

PERFORMANCE ANALYSIS 99

-5 -4 -3 -2 -1 0 1 2 3 4 510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Pro

b. o

f Occ

urre

nce

Normalized residuals

Figure 6.1 Histogram of normalized residuals with single cross-validation

We see from Figure 6.1 that the first requirement is met. None of the normalized residuals

is above K (in fact they are well below). The second requirement can be checked by

plotting a quantile-quantile plot which is shown in Figure 6.2.


-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

Standard Normal Quantiles

Qua

ntile

s of

nor

mal

ized

resi

dual

s

Figure 6.2: Q-Q plot of the normalized residuals

We see that the empirical quantile corresponding to the normalized residuals is below the

corresponding unit Gaussian quantile (blue line), except for a few quantiles for which it is

still very close. Upon closer examination of the large normalized residuals, we find that

they come from data on April 6, 2000, July 16, 2000 and March 31, 2001 at locations

where the test measurement was poorly surrounded by measurements, and a large gradient

is affecting the test measurement but not the remaining measurements. It is not a

consequence of the randomness of the residual field but rather of a large undetected feature.

It is in this type of situation that the threat model recognizes a poorly sampled location and

inflates the error bound according to what has been observed. As a result, these large

residuals would be mitigated in the real system.

PERFORMANCE ANALYSIS 1016.1.2 DATA DEPRIVATION

In this subsection, we apply data deprivation. That is, instead of only excluding the data to

be estimated, we exclude a whole set. As we said earlier, ideally the algorithm should be

able to handle any deprivation scheme. But we are limited by our assumption about

stationarity. Situations like the one pictured in Figure 4.11 are likely to cause problems

when we suppose that all the IPPs below 25 degrees of latitude are not observed. There are

many possible data deprivation schemes. Here, we choose the deprivation scheme that best

simulates a user: we exclude all the data coming from the station where the measurement

was taken. This method more closely mimics the estimation error of a user, because there

is no ionospheric information in similar lines of sight as the one tested. Figure 6.3 shows

the empirical distribution of the normalized residuals.

-5 -4 -3 -2 -1 0 1 2 3 4 510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Pro

b. o

f Occ

urre

nce

Normalized residuals

Figure 6.3 Histogram of normalized residuals with station removal

PERFORMANCE ANALYSIS 102Again, there is no residual above 5.33. The fact that the residuals have not increased with

less data indicates that the algorithm is taking into account correctly the smaller data set

and the increasing confidence bounds. Figure 6.4 shows the quantile-quantile plot of the

residuals. Again, the plot is similar to the previous q-q plot.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

Standard Normal Quantiles

Qua

ntile

s of

nor

mal

ized

resi

dual

s

Figure 6.4: Q-Q plot of the normalized residuals with station removal

6.2 PERFORMANCE METRICS

Once we are confident that the designed algorithm is safe, we want to know how small the

calculated error bounds are and how this translates to the user’s bound on inaccuracy. In

this section we explain the different ways of measuring the performance of an ionosphere

PERFORMANCE ANALYSIS 103estimation algorithm. Before introducing the two metrics used, we need to present the

simulation tool used to generate the results.

6.2.1 MAAST

The MAAST is a service volume analysis tool. The goal of this tool is to predict the

availability of WAAS. For this purpose, it computes all the configurations over a period of

time (typically 24 hours) long enough to be representative of all typical satellite

configurations. For each one MAAST computes the message that would be sent to the

users, in particular, the UDRE and the GIVE. After that, it computes the Vertical

Protection Level that a user would compute for different locations over CONUS. A full

description of MAAST can be found in [Jan].

All terms except the GIVE depend only on the geometry of the satellites (and the history of

the geometry). As shown, the GIVE is dependent on the actual behavior of the ionosphere

through the inflation factor u0. MAAST has two ways of modeling the effect of the

inflation factor on the GIVE, one with and the other without real data. The GIVE was

computed using the process illustrated in Figure 5.6.

Without real data, we can evaluate the performance of the algorithm when the ionosphere is

described conservatively by the nominal model. Assuming this model, we know that the

chi-square statistic will be above a threshold, T, with probability Pfa (probability of false

alarm). If we set this probability low enough, we know that u02 will be below αT with high

probability. Let us take Pfa=10-4. For a given day with nominal ionospheric conditions we

have an upper bound of u0 for any situation. There are two main drawbacks in this option.

First, it is slightly pessimistic in the sense that the chi-square statistic is usually much lower

than the threshold by a factor of two or three. Second, it does not allow us to simulate real

situations where the ionosphere is slightly disturbed.

MAAST can also be used with archived real data. Ideally, we would use archived data that

replicates exactly the data that is fed to the estimation algorithm in real time. As said

earlier, this data is noisier than the post-processed data. Unfortunately, access to the rea;

data being fed to the real time algorithm (the noisy data) is limited. The idea is to use the

PERFORMANCE ANALYSIS 104post-processed data. There is a large amount of it and it covers a broad range of

ionospheric behavior. Because it has low measurement noise, as opposed to the real

situation, we need to compute the chi-square as if there was no measurement noise. If we

did not do this, the chi-square statistic would be lower than it should be. However, we still

consider that there is measurement noise when computing the estimation variance. It is for

this reason that in this thesis real data is used. This method will allow us to more

accurately evaluate the inflation factor during nominal and disturbed ionospheric behavior.

The drawback is that the post-processed data filters out a large amount of measurements

because their noise it too large. As a consequence, the data appears to be scarce in regions

where it would not be in real time.

6.2.2 GRID IONOSPHERIC VERTICAL ERROR AND VPL

The Grid Ionospheric Vertical Error is the quantity that is sent to the user at each IGP. For

historical reasons, the GIVE is defined as:

3.29 GIVEGIVE σ=

The first and most direct way of evaluating the performance of the algorithm is by

computing the error bounds, or equivalently, the Grid Ionospheric Vertical Error over a

period of time. This period of time should be such that the set of satellite configurations is

representative of any situation. Since the period of a satellite is approximately 12 hours, a

period of 24 hours is enough. The next question is how to display the results. It is not

enough to compute the mean of the error bounds and see how this mean compares to the

mean of another estimation algorithm. It could be that the error bounds are larger where

we do not really care but smaller where we want them to be small. Showing the mean of

the error bound per location is still not enough because the extreme values may be averaged

out and it is those extreme values that can hurt availability. Figure 6.5 is an example of

how we will plot the GIVE values over CONUS. As indicated earlier, the circles

correspond to the location of the IGPs.


GIV

E (m

) - 9

5%

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

3.6

4.5

6

15

45

NM

-180 -160 -140 -120 -100 -80 -6010

20

30

40

50

60

70

Longitude (deg)

Latit

ude

(deg

)

GIVE values

Figure 6.5: GIVE map over CONUS

To obtain this plot we first run MAAST. At each location we can create a histogram of the

GIVE on the grid. At each IGP, the color code corresponds to the magnitude of the 95th

percentile of the GIVE. Similarly, we can plot VPL quantiles, where the map gives the

position of the user.

6.3 RESULTS WITHOUT THREAT MODEL

Here we show the results without including the threat model term. These represent an

upper bound of the performance. We will use here real data from a quiet day (July 2,2000)

PERFORMANCE ANALYSIS 106and a disturbed day (September 8,2002). The threat model is not used, and the storm

detector is turned off.

In order to have an idea of the performance of the algorithm designed in this thesis, we will

compare the results given by MAAST with kriging with the results given by the proposed

Final Operation Capability (FOC) estimation algorithm for WAAS proposed by Raytheon.

This other algorithm has several common features with the algorithm proposed in this

thesis [Cormier]. The Raytheon proposed algorithm is based on a planar fit and a dynamic

inflation factor of the GIVE based on the chi-square statistic.

6.3.1 QUIET DAY

GIV

E (m

) - 9

5%

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

3.6

4.5

6

15

45

NM

-180 -160 -140 -120 -100 -80 -6010

20

30

40

50

60

70

Longitude (deg)

Latit

ude

(deg

)

GIVE values

Figure 6.6: GIVE map for July 2, 2000 using kriging

PERFORMANCE ANALYSIS 107In Figure 6.6, the 95 percentile of the GIVE at each location is plotted. These GIVEs

combined with the other errors (in particular the UDRE) as explained in Chapter 1 produce

the VPLs plotted in Figure 6.7. This map can be compared to the VPL produced by the

Raytheon proposed FOC algorithm and plotted in Figure 6.8. We see that the region with

VPLs below 20 meters has increased considerably. This will not increase the level of

service immediately because the critical VAL is 50 meters, which is already attained on

quiet days with the current algorithm. It would however put WAAS closer to the next level

of service, which requires VPLs below 20 meters 99.9% of the time. Also, it would make

it more robust to satellite outages.

VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)

VPL as a function of user location

Figure 6.7: VPL map for July 2, 2000 using kriging


VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.8: VPL map for July 2, 2000 using Raytheon’s algorithm

6.3.2 DISTURBED DAY

In this section we examine the performance of the algorithm during a mildly disturbed day

which did, however, considerably limit WAAS performance. The same settings as in the

previous section were used. As expected, the GIVE values (Figure 6.9) are larger than on a

quiet day. This is due to the larger decorrelation of the ionosphere, which is captured in the

algorithm through the inflation factor u02, whose determination is the subject of Chapter 4.


GIV

E (m

) - 9

5%

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

3.6

4.5

6

15

45

NM

-180 -160 -140 -120 -100 -80 -6010

20

30

40

50

60

70

Longitude (deg)

Latit

ude

(deg

)

GIVE values

Figure 6.9: GIVE map for September 8, 2002 using kriging

Figure 6.10 shows the corresponding VPLs, which are larger than on a quiet day and Figure

6.11, the VPLs obtained with the Raytheon proposed FOC algorithm. The difference now

is dramatic: the VPLs have been reduced by almost 40%.


VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.10: VPL map for September 8, 2002 using kriging

VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.11: VPL map for September 8, 2002 using Raytheon’s algorithm

PERFORMANCE ANALYSIS 111This difference in performance comes mainly from the different characterizations of the

ionosphere. In both algorithms, the ionosphere is characterized as a planar trend with a

residual random field (see Chapter 3). The difference is that in our algorithm we assume

that the residual field has a distance dependent covariance, that is, IPPs that are closer are

more likely to be correlated, whereas in the other algorithm, the residual field is

uncorrelated from one point to another. While that might be a good approximation on quiet

days (although there is already a degradation during quiet days as seen in the previous

subsections) it does not work as well during mildly disturbed days, when the ionosphere,

though not clearly following a planar trend, is still correlated with distance.

6.4 RESULTS WITH THREAT MODEL

In this section we show the results including the threat model term. As said in Section 4.4,

the threat model term is dependent on the algorithm. A different threat model is used for

the algorithm based on kriging and the algorithm used for comparison. We will show the

same plots as before where we have included the threat model term.

As mentioned earlier, the threat model is meant to protect users against large ionospheric

gradients. For this reason, the GIVEs are mainly going to be degraded at the edge of

coverage.

6.4.1 QUIET DAY

Figure 6.12 shows how Figure 6.6 is modified once we include the threat model term.

Now, the region with very low GIVEs is smaller but still covers most of the CONUS

region.


GIV

E (m

) - 9

5%

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

3.6

4.5

6

15

45

NM

-180 -160 -140 -120 -100 -80 -6010

20

30

40

50

60

70

Longitude (deg)

Latit

ude

(deg

)

GIVE values

Figure 6.12: GIVE map for July 2, 2000 using kriging with threat model

In Figure 6.13 we show the corresponding VPL map. As expected, the performance is

degraded, but most of CONUS is still below 20 meters VPL 95% of the time. The

performance of the Raytheon proposed FOC algorithm (Figure 6.14) is also degraded such

that the relative improvement is similar to the previous one.


VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.13: VPL map for July 2, 2000 using kriging with threat model

VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.14: VPL map for July 2, 2000 using Raytheon’s algorithm with threat model

PERFORMANCE ANALYSIS 1146.4.2 DISTURBED DAY

In this subsection we have added the threat model term in the GIVE on September 8, 2002.

We see again in Figure 6.15 that the GIVE is degraded on the edge of coverage. The effect

is not as large as for a quiet day due to the quantization of the GIVE, which is much wider

at larger values. The effect on the VPL is shown in Figure 6.16. Again the performance is

slightly degraded compared to the case without the threat model. As a comparison, we plot

the VPL for the Raytheon proposed FOC algorithm with threat model in Figure 6.17.

GIV

E (m

) - 9

5%0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

3.6

4.5

6

15

45

NM

-180 -160 -140 -120 -100 -80 -6010

20

30

40

50

60

70

Longitude (deg)

Latit

ude

(deg

)

GIVE values

Figure 6.15: GIVE map for September 8, 2002 using kriging


VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.16: VPL map for September 8, 2002 using kriging

VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.17: VPL map for September 8, 2002 using Raytheon’s algorithm

PERFORMANCE ANALYSIS 1166.5 LOSS OF AVAILABILITY DUE TO THE MESSAGE

At the end of Chapter 4 we ended up with an ionospheric estimation algorithm that could

work if the user had all the measurement information. However, at that stage the algorithm

was not useful with the current WAAS standards. The algorithm was modified in Chapter

5 to take into account the current WAAS standards. There are two steps involving

bandwidth limitations that degrade the performance that are due to bandwidth limitations:

the use of the ionospheric grid, which requires an additional inflation of the error bound (as

we saw in Chapter 5) and the quantization of the Grid Ionospheric Vertical Error (instead

of sending the actual value, the master station sends an index corresponding to a tabulated

value). In the current algorithm, the quantization is indicated in the color bar of Figure 6.5.

In this section, we evaluate the performance of the algorithm without bandwidth

limitations. Figure 6.18 shows the VPL map if we suppose that there is no quantization

and there is no degradation due to the grid.

VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.18: VPL map for July 2, 2000 using kriging without bandwidth limitations

PERFORMANCE ANALYSIS 117This map can be compared with the VPL map shown in Figure 6.13. We see that we gain

some performance, but the difference is small. To know whether the (small) performance

loss of the full algorithm is due to the quantization or the inflation due to the grid, we plot

in Figure 6:19, the VPL without quantization but with the grid.

VPL (m) - 95%< 20 < 30 < 40 < 50 < 60 < 80 < 100 < 120 > 120

-120 -110 -100 -90 -80 -7020

25

30

35

40

45

50

Longitude (deg)

Latit

ude

(deg

)


Figure 6.19: VPL map for July 2, 2000 using kriging without quantization

This map is virtually identical to the previous one, so the grid does not introduce a

significant degradation.

Chapter 7

Conclusion

7.1 SUMMARY OF RESULTS

The goal of this thesis was to design an algorithm for satellite based augmentation systems

that, from GPS measurements collected in a network of reference stations, estimates the

delay caused by the ionosphere at any location within the reference stations, and provides

an error bound. Several problems needed to be addressed. First, the measurements are

irregularly scattered in the region to be estimated. Second, the measurements are noisy.

Third, the process noise due to the ionosphere cannot be characterized by a single process

noise and must be bounded in real time. Finally, for such an algorithm to be useful, it has

to fit within the current standards of satellite based augmentation systems. We review now

the different contributions of this thesis to solve this problem.

7.1.1 VERTICAL IONOSPHERIC DELAY STRUCTURE

The first step (Chapter 3) addressed the characterization of the vertical ionospheric delay,

that is, the ionospheric delay projected on the thin shell. We showed that it is possible to

interpret the vertical ionospheric delay as the combination of a planar trend and a random

residual field correlated with distance. This residual field can be characterized by a

variogram with a non-zero intercept and a certain slope. We determined the variogram for

CONCLUSION 119a quiet day (or nominal day) from real data, and showed that for more disturbed

ionospheric behavior we could assume that the nominal variogram multiplied by a certain

factor, u, can be used. Because this model assumes stationary stochastic processes it

cannot capture correctly the occasional non-stationarity of the ionosphere. In particular, we

noted that large gradients pose a very real threat.

7.1.2 IONOSPHERIC ESTIMATION

An ionospheric estimation algorithm was designed in Chapter 4 based on the

characterization given in Chapter 3. The algorithm is based on the well known estimation

method called kriging, which is a type of minimum mean square estimator. The classical

application of kriging requires the knowledge of the process noise covariance. Since the

process noise covariance is not known in our problem, a covariance that conservatively

describes the real covariance needed to be estimated. This was realized by inflating the

nominal covariance (given in Chapter 3) by a factor dependent on the number of

measurements, a chi-square statistic computed from the measurements and the

measurement noise. The dependence on measurement noise results in the error bound

being largely dependent on the quality of the measurements. This is a different situation to

one where the process noise is well known (in this last case, the measurement noise has

little impact on the final error bound because it is averaged out), and the successful

treatment of this problem in this thesis might be its most relevant contribution. (To fully

protect the user, the algorithm includes a term coming from the threat model, an external

part of the algorithm not developed in this thesis. The threat model protects users against

severe gradients and non-stationarity.)

Another contribution of this thesis is the adaptation of kriging to the WAAS ionospheric

grid. Typically, kriging requires the user to know all measurements to compute the

estimate and the error bound. We have shown that the user can interpolate the values

broadcast by WAAS in the ionospheric grid (computed by the master station using kriging)

if the error bounds at the grid points are carefully computed. This was done in Chapter 5

using the specific properties of the covariance and the interpolation scheme used by the

receivers (and specified in the MOPS).

CONCLUSION 1207.1.3 PERFORMANCE

In Chapter 6 we showed that the algorithm designed in this thesis can provide a significant

improvement compared to the current design. It is difficult to give a single number

reflecting the improvement; the maps shown in Chapter 6 might summarize the situation

better. Additionally, the relative improvement depends on the day that is considered. We

have shown that on a quiet day there is a sizable relative improvement: the region with a

VPL lower than 20 meters covers most of CONUS. However, this improvement does not

have an impact on availability on a quiet day, since the availability is already maximal

(although it does bring the system closer to the next level of performance). The

improvement has much more effect on mildly disturbed days, and the plots shown in

Chapter 6 corresponding to September 8, 2002 suggest that the VPLs could be lowered as

much as $0% with this algorithm, bringing VPLs that were as high as 60 meters to 40

meters. As an average over several days, we expect this algorithm to provide at least a

15% to 20% reduction in VPL.

7.2 SUGGESTIONS FOR FUTURE WORK

The threat model has not been covered in this thesis. We have seen that there is a large

difference between performance with and without the threat model. That means that our

model is not accounting correctly for the violations of stationarity. This is not unexpected

as we are using stationary random models. An ideal estimation algorithm would close the

gap between the results with threat model and without; in other words, the threat model

would not be needed. Future work should focus on integrating explicitly the possible

threats in the ionospheric model, instead of relying on the threat model. (This was one of

the goals of kriging, since kriging recognizes undersampled situations , but was not

achieved because the threats turn out to be much more catastrophic).

Another area of future research is the adaptation of this algorithm to equatorial latitudes.

The algorithm presented here was designed for mid-latitude regions (all data used comes

CONCLUSION 121from CONUS), where the ionosphere is well behaved most of the time, and where the thin

shell model is a surprisingly good approximation. This is not the case in equatorial regions.

There, the ionosphere has much more structure and variability, and the thin shell model

does not perform as well [Komjhaty]. Although much of the analysis done in this thesis

relies on the thin shell model, it is possible to relax it to improve the performance in

equatorial regions. For example, one could assume that instead of having all the electron

content concentrated at one height, we could assume a certain (fixed) vertical profile. This

way, we would preserve the two-dimensionality of the estimation without ignoring the

three-dimensional character of the ionosphere.

Appendix A

COMPLEMENTARY DEFINITIONS AND PROOFS

This appendix contains the definition of the Gaussian overbound used in this thesis, the

proof of inequality (4.30) (which is the basis for the computation of parameter β) and the

expression of the constant ε introduced in Chapter 5.

A.1 GAUSSIAN OVERBOUND

Let us consider a zero mean distribution over the real axis, p(x). We will say that q(x) is an

upper bound of p(x) if we have for any s:

( ) ( )0 0

s s

p x dx q x dx≥∫ ∫

If the distribution q is Gaussian, we say we have a Gaussian overbound. This definition of

overbound is interesting because of the result shown in [DeCleene], which states that if the

distributions are unimodal and symmetric the convolution of the overbounds of two

distributions is an overbound of the convolution of the two original distributions. This

result allows us to use covariance propagation to compute probabilities without strictly

having Gaussian distributions.

APPENDIX A 123A.2 INEQUALITY IN SECTION 4.2

In this section we show a proof of the inequality (4.2) which is:

( )1

1 1

1 1

pp pp

k kk k

b b= =

⎛ ⎞⎛ ⎞⎜ ⎟+ ≤⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎝ ⎠∏ ∏ +

It might be possible to derive this inequality from classic inequalities. We give here a short

proof. It is easy to check that the function:

( )ln 1 xe+

is convex. As a consequence we have the inequality

( )1

1

1

1ln 1 ln 1

p

kk k

pap a

k

e ep

=

=

⎛ ⎞∑⎜ ⎟+ ≤ +⎜ ⎟⎝ ⎠

∑

We obtain our inequality by doing the change of variable kakb e= .

A.3 BOUNDING ADDITIONAL TERM IN GIVE

In section 5.3.4 we needed a positive constant such that:

( ) ( )0 ' , ,k kC x x C x x ε≤ − ≤

for any x within the quadrant we are trying to protect. ( )' , kC x x is a concave function that

is an upper bound of the covariance and is as close to the covariance as possible. The

expression for the covariance is:

APPENDIX A 124

( ) 20,

kx xa

kC x x u ce−

−=

Let us define:

{ }{ }

min

max

min | is in quadrant

max | is in quadrantk

k

d x x x

d x x x

= −

= −

We define C’ as:

( )max maxmin min2

0max min

max min

' ,d dd d

a a ak k

u cC x x d e d e x x e ed d

− − − −⎛ ⎞⎛ ⎞= − − −⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠

a−

We want to show that there exists ε such that for any kd x x= − where x is in the quadrant

we have:

max maxmin min220

max min 0max min

0d dd d d

a a a a au c d e d e d e e u ced d

ε− − − − −⎛ ⎞⎛ ⎞

≤ − − − −⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠≤

The first inequality is easy to verify because C’ is affine, coinciding with C at dmin and dmax

and C is a convex function of d (but not of x). Let us now define mind dδ = − and

max max mind dδ = − . The expression above is now:

maxmin20

max

0 1 1d

a au ce e eδ δδ a ε

δ− −⎛ ⎞⎛ ⎞⎛ ⎞

≤ − − −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

−≤

We now find the maximum of the function:

( )max

max

1 1 a af e eδ δδδ

δ− −⎛ ⎞⎛ ⎞

= − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

over [0,δ]. The maximum is attained when:

APPENDIX A 125max

max

ln 1 aa ea

δδδ

−⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

We end up with:

( )max max max

max max

max

max max

max

max

1 ln 1 1 1

1 1 ln 1 1

a a

a a

a af e ea

a e ea

δ δ

δ δ

δδδ δ

δδ

− − −

− −

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞≤ + − − − −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞= + − − −⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

aeδ

This is a function of the parameter a (a parameter defining the variogram which is set to

32000km (see Chapter 3)) and maxδ which in the 5 by 5 degree cells will typically be

smaller than 700km. With these constants we end up with:

( ) 41.25*10f δ −≤

As we saw, c=2 so:

2 40 2.5*10 muε −≤ 2

This term will result in a very small contribution to the GIVE, and could potentially be

ignored.

A.4 CORRECTION TO THE PHMI EQUATION

This section was added after the submission of this work. The changes introduced here

have a very limited impact on the final GIVE. This note is included here for completeness

purposes.

APPENDIX A 126Equation 4.19 assumes that the estimation error is independent of the chi-square statistic.

In the general case, they are not independent. This note provides an upper bound of the

true PHMI as a function of the expression provided in Equation 4.19.

x is the estimation error: T

meas unknownx I Iλ= −

y is the vector of reduced measurements and has the covariance S(u), which will be

shortened as S.

The covariance of [x y]T is given by:

[ ]( )covT

T sx y

Sρ

ρ⎡ ⎤

= ⎢ ⎥⎣ ⎦

Where we have:

( ) ( )( )( )

2 2 2

20

20

process meas

T

v

s u

S u C M

u C M C

σ λ σ λ

ρ λ

= +

= Γ + Γ

= Γ + −Γ

Under our model, P(HMI|u) is given by:

( ) ( )( ) ( )2 2

| ,2 2 2

P HMI u p x y dxdyTK y y K xprocess meas

ρ

α σ λ σ λ

=⎛ ⎞ + ≤⎜ ⎟⎝ ⎠

∫

The density is indexed by the correlation between X and Y.

We now do the change of variables: 12

12

0

0

X s xY y

S

−

−

⎡ ⎤⎡ ⎤ ⎡⎢ ⎥=

⎤⎢ ⎥ ⎢⎢ ⎥ ⎥⎣ ⎦ ⎣⎢ ⎥⎣ ⎦

⎦

The expression for the PHMI is now:

APPENDIX A 127

( ) ( )( ) ( )

( )

2 2

| ,2 2 2

,2

P HMI u p X Y dXdYTK Y SY K sXprocess meas

p X Y dXdYTY Y sX

ρ

ρ

α σ λ σ λ

=⎛ ⎞ + ≤⎜ ⎟⎝ ⎠

=

Σ + ≤

∫

∫

Where we have:

( )2 2K Sprocessα σ λ⎛ ⎞Σ = ⎜ ⎟⎝ ⎠

( )2 2measK σ λ=

And the covariance of [X Y] is given by:

3

1cov

T

bn

X bY b I −

⎛ ⎞ ⎡ ⎤⎡ ⎤= = ∆⎜ ⎟ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦⎝ ⎠

where: 1 12 2b S s ρ

− −=

Notice that we always have 1T

b b < , because the matrix above is a covariance matrix.

We now prove that:

( ) ( )01, ,

12 2T

p X Y dXdY p X Y dXdYb bT TY Y sX Y Y sX

ρ ρ=≤−

Σ + ≤ Σ + ≤∫ ∫

We have:

( ) ( ), ,2 2

X

X

p X Y dXdY p X Y dY dXT TY Y sX Y Y sX

ρ ρ

=+∞

=−∞

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟Σ + ≤ Σ + ≤⎝ ⎠

∫ ∫ ∫

We now examine the integral over Y:

( )( )

11122

22

1,22 2

TT TbX Y X Y

bnp X Y dY e dYT TY Y sX Y Y sX

ρ

π

−⎡ ⎤ ⎡ ⎤− ∆− ⎣ ⎦⎣ ⎦−= ∆

Σ + ≤ Σ + ≤∫ ∫

APPENDIX A 128We have:

1 Tb b b∆ = −

We develop the PHMI expression:

( )

( )

1

22

1122

22

1 2 12 1 1 1

22

1

2 2

1

2 1 2

TT Tb

T TT T

n T T T

X Y X Y

bn

bb b b bY I Y Y X Xb b b b b b

nT

e dYTY Y sX

e db b TY Y sX

π

π

−

−

⎡ ⎤ ⎡ ⎤− ∆− ⎣ ⎦⎣ ⎦−

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− + − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− − −⎝ ⎠ ⎝ ⎠⎝ ⎠

−

∆

Σ + ≤

=−

Y

Σ ≤ −

∫

∫

Let us now do the change of variable: 1

3 1 1

T

n T T

bb bZ Y I X Yb b b b

−

−

⎛ ⎞= − + = −⎜ ⎟− −⎝ ⎠

c

( )

( ) ( ) ( )

22

22

1 2 12 1 1 1

22

12 1

22

1

2 1 2

1

2 1 2

T TT T

n T T T

TT

n T

bb b b bY I Y Y X Xb b b b b b

nT

bbZ I Z Xb b

nT

e db b TY Y sX

e db b TZ c Z c sX

π

π

−

−

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− + − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− − −⎝ ⎠ ⎝ ⎠⎝ ⎠

−

⎛ ⎞⎛ ⎞⎜ ⎟− + +⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠

−

− Σ ≤ −

=− + Σ + ≤ −

∫

∫

Y

Z

For this step of the calculation, we have used the fact that:

( )1

3 11

TT

n T

bbI b b bb b

−

−

⎛ ⎞+ = −⎜ ⎟−⎝ ⎠

b

which implies that:

( ) ( )1

3 1 11 1

T T

n T TT T

b bb bIb b b bb b b b

−

−

⎛ ⎞+ =⎜ ⎟− −− −⎝ ⎠

Tb b

So far we have only proceeded by equalities, we now start taking upper bounds. The first

upper bound is:

APPENDIX A 129

( ) ( )

2 22 2

1 12 21 1

22

T TT T

n nT Tbb bbZ I Z X Z I Z X

b b b be dZ eT TZ Z sXZ c Z c sX

− −

⎛ ⎞ ⎛⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜− + + − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜− −⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝≤

Σ ≤ −+ Σ + ≤ −∫ ∫ dZ

⎞⎟⎟⎠

This step can be easily proven and is very conservative.

We can also write:

( )2

221 12 1 2

2 2

TT

Tn TbbZ I Z X Z Z Xb be dZ e

T TdZ

Z Z sX Z Z sX

−

⎛ ⎞⎛ ⎞⎜ ⎟− + +⎜ ⎟⎜ ⎟ − +⎜ ⎟−⎝ ⎠⎝ ⎠ ≤

Σ ≤ − Σ ≤ −∫ ∫

This proves the inequality (1.1).

( ) ( )01, ,

12 2T

p X Y dXdY p X Y dXdYb bT TY Y sX Y Y sX

ρ ρ=≤−

Σ + ≤ Σ + ≤∫ ∫

This equation shows that the PHMI is 11 Tb b−

larger than the one predicted by Equation

4.19.

We now provide an overbound of the parameter 11 T

Bb b

=−

that can be studied offline.

The expression for in the case of the planar fit for a given u is: Tb b

( )( ) ( )( ) ( )( )( )

12 2 2 21

0 0 0

2 2 2 20 2

T T T TTv vT

T Tv decorr

u C M u C u C M u C M u CSb bs u C M u C u

λ λρ ρλ λ λ σ

−− + − Γ Γ + Γ Γ + −

= =+ − +

2

We now look for a bound on this expression for u between u=1 and infinity. To do this, we

are going to use the fact that:

( )( )0 0vC M CλΓ + − =

APPENDIX A 130

This allows us to write that:

( ) ( ) ( )( ) ( )( )

122 20 0

2 2 2 20

1

2

T T T Tv vT

T Tv decorr

u C C u C M Cb b

u C M u C u

λ λ

λ λ λ σ

−− − Γ Γ + Γ Γ −

=+ − +

0 C

As a result, we have:

( ) ( )( ) ( )( )

14 2

0 0

2 2 2 20 2

T T T Tv vT

T Tv decorr

u C C u C M C Cb b

u C M u C u

λ λ

λ λ λ σ

−− Γ Γ + Γ Γ −

≤+ − +

0

0 vλ

We also have:

( ) ( )( ) ( ) ( ) ( )( ) ( )1 1

2 20 0 0 0 0

T T T T T T T Tv v vC C u C M C C C C u C C Cλ λ λ

− −− Γ Γ + Γ Γ − ≤ − Γ Γ Γ Γ −

and:

( ) ( )2 2 2 2 2 2 2 20 02 2T T T T

v decorr v decorru C M u C u u C u C uλ λ λ σ λ λ λ+ − + ≥ − + σ

Finally:

( ) ( ) ( )1

0 0 02

0 2

T T T Tv vTT T

v decorr

C C C C Cb b

C Cλ λ

λ λ λ σ

−− Γ Γ Γ Γ −

≤− +

This expression is now independent of u. It is now possible to study the actual distribution

of B and set the αn accordingly, such that the true PHMI is below the allocation. It is

expected that this correction will have a minimal impact on the inflation factor.

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